[OAI-general] math>Number Theory>Fermat's Last Theorem
kerryevans evans
kerryme1165@hotmail.com
Sat, 08 Dec 2001 16:25:30 +0000
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<P>The attached is an amateur attempt at proof of Fermat's Last Theorem. It requires only the most basic knowledge of Number Theory. This probably more properly is under the domain of Algebra since it addresses the most fundamental properties of equations.</P></DIV>
<P> Kerry Evans</P></DIV>
<P> kerryme1165.hotmail.com</P>
<DIV></DIV>
<P><BR><BR> </P> <BR><BR><BR>
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<p class=MsoNormal style='text-align:justify;text-indent:.5in'><span
style='mso-tab-count:1'> </span><span style='mso-tab-count:1'> </span><span
style="mso-spacerun: yes"> </span>IN DEFENSE OF MR. FERMAT</p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'>During the course of studies on
the Goldbach<span style="mso-spacerun: yes"> </span>Conjecture, using finite
methods, what seems to be an elementary proof of Fermat’s “Last” Theorem has
been found.<span style="mso-spacerun: yes"> </span>Astonishing here is the lucidity
of the arguments and immediacy of their logic.<span style="mso-spacerun: yes">
</span>Hopefully, by (numeric) application to the so-called “hard” problems of
Number Theory, some manner of agreement (disputation) will arise.</p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:1'> </span><span style="mso-spacerun:
yes"> </span>FERMAT’S LAST THEOREM</p>
<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes"> </span><span style='mso-tab-count:1'> </span>Suppose the
following equation has the solution for positive r, a and b.</p>
<p class=MsoNormal style='text-align:justify'><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span
style="mso-spacerun: yes"> </span>(1)<span style="mso-spacerun: yes">
</span>r^n = a^n + b^n<span style="mso-spacerun: yes"> </span>where a>0
and b>0.</p>
<p class=MsoBodyText><span style='mso-tab-count:1'> </span><span
style='mso-tab-count:1'> </span>Clearly the following congruences
must hold (implied).</p>
<p class=MsoNormal style='text-align:justify'><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span
style="mso-spacerun: yes"> </span>(2)<span style="mso-spacerun: yes">
</span>r^n -<span style="mso-spacerun: yes"> </span>b^n = 0<span
style="mso-spacerun: yes"> </span>(mod<span style="mso-spacerun: yes">
</span>a^n)</p>
<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes"> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span
style="mso-spacerun: yes"> </span>(2)’ r^n – a^n = 0<span style="mso-spacerun:
yes"> </span>(mod<span style="mso-spacerun: yes"> </span>b^n)</p>
<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes"> </span></p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify'><span
style="mso-spacerun: yes"> </span>From now on forsaking implicit-only
relations (2) and (2)’ can be used to further specify the consequences of (1),
using the single valued function,F(x,y)</p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><b><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style="mso-spacerun: yes"> </span>F(x,y)<o:p></o:p></b></p>
<p class=MsoBodyText style='margin-left:15.0pt;text-indent:21.0pt'>Let {c}
represent, i.) c.GE. 0,<span style="mso-spacerun: yes"> </span>the greatest
Integer in c, or ii.) c< 0,. -({|c|} + 1).</p>
<p class=MsoBodyText align=left style='margin-left:15.0pt;text-align:left'><i><span
style="mso-spacerun:
yes"> </span></i>F(x,y)
is Defined<span style="mso-spacerun:
yes">
</span>y: y.NE.0,<span style="mso-spacerun: yes"> </span>F(x,y) = x – ( |y| *
{ x / |y| }),<span style="mso-spacerun: yes"> </span>y: y=0<span
style="mso-spacerun: yes"> </span>F(x,y) is undefined.<span
style="mso-spacerun: yes"> </span>F may be referred to as the “least positive
(or 0) remainder of x on division by |y| function.”</p>
<p class=MsoBodyTextIndent>Explicitly now as [(1) implies (2)] and [(1) implies
(2)’] when [r,a,and b] are positive,<span style="mso-spacerun: yes">
</span>(3)<span style="mso-spacerun: yes"> </span>F((r^n – b^n), a^n) =
0<span style="mso-spacerun: yes"> </span>as well as (3)’ F((r^n,a^n), b^n) =
0.</p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span
style="mso-spacerun: yes"> </span>ORDER</p>
<p class=MsoNormal style='margin-left:15.0pt;text-align:justify;text-indent:
21.0pt'>The following “order” relations are definitive of the assumption [a
> b] when [r, a and b]<span style="mso-spacerun: yes"> </span>are positive.</p>
<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes"> </span>(4)<span style="mso-spacerun: yes"> </span>F(r^n,a^n) –
b^n = 0<span style="mso-spacerun: yes"> </span></p>
<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes"> </span>(4)’ F(r^n,b^n) – a^n .NE. 0<span style="mso-spacerun:
yes"> </span><span style="mso-spacerun:
yes"> </span></p>
<p class=MsoNormal style='margin-top:12.0pt'><span style="mso-spacerun:
yes">
</span>PROOF</p>
<p class=MsoNormal style='margin-top:12.0pt;margin-right:0in;margin-bottom:
0in;margin-left:.5in;margin-bottom:.0001pt;line-height:12.0pt'>Let [r,a,and b]
be positive.<span style="mso-spacerun: yes"> </span></p>
<p class=MsoNormal style='margin-left:15.0pt;text-align:justify'><span
style="mso-spacerun: yes"> </span>First suppose a^n = b^n.Then, (1) can be
represented r^n = 2*(a^n) which has no solution in positive in positive
Integers</p>
<p class=MsoNormal style='margin-left:15.0pt;text-align:justify;text-indent:
21.0pt'>Let [a^n > b^n].<span style="mso-spacerun: yes"> </span>Then
F(r^n,a^n) =<span style="mso-spacerun: yes"> </span>b^n<span
style="mso-spacerun: yes"> </span>and (4) immediately follows.<span
style="mso-spacerun: yes"> </span>Regarding<span style="mso-spacerun: yes">
</span>(4)’,</p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify'><span
style="mso-spacerun: yes"> </span>F(r^n,b^n) =
F(a^n,b^n)<span style="mso-spacerun: yes"> </span>.NE.<span
style="mso-spacerun: yes"> </span>a^n<span style="mso-spacerun: yes">
</span>so that:<span style="mso-spacerun:
yes">
</span></p>
<p class=MsoNormal style='margin-top:12.0pt;text-align:justify'><span
style='font-size:13.0pt;mso-bidi-font-size:12.0pt'><span style="mso-spacerun:
yes"> </span><span style="mso-spacerun:
yes"> </span>F(r^n,b^n) – a^n = F(r^n,b^n) – F(a^n,b^n) –
d*(b^n)<span style="mso-spacerun: yes"> </span>.NE. 0 <o:p></o:p></span></p>
<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes">
</span>where d .GE.1.</p>
<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun: yes">
</span></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='margin-left:15.0pt;text-align:justify;text-indent:
21.0pt'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify;text-indent:.5in'>Conversely,
suppose [a^n is not greater than b^n] while (4) holds.<span
style="mso-spacerun: yes"> </span>Then [b^n > a^n] and transformation
of<span style="mso-spacerun: yes"> </span>a to b<span style="mso-spacerun:
yes"> </span>and<span style="mso-spacerun: yes"> </span>b to a<span
style="mso-spacerun: yes"> </span>has no effect on (1)<span
style="mso-spacerun: yes"> </span>but “rectifies” the representations of (4)
and (4)’ accordingly as [r><u>b>a</u>>0].<span style="mso-spacerun:
yes"> </span>Thus, (4)’ rightfully becomes, F(r^n,a^n) – b^n<span
style="mso-spacerun: yes"> </span>.NE.<span style="mso-spacerun: yes">
</span>0.<b>!</b> which is inconsistent with prior (4).<span
style="mso-spacerun: yes"> </span>Proof of the case (4)’ follows
complementarily. <b>!</b> denotes contradiction. Thus, a^n > b^n may be
defined in this manner.</p>
<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes"> </span></p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify'><span
style="mso-spacerun: yes"> </span>Prominent now becomes the anticipation
that (1) has a distinct ordering of a^n and b^n is contradictory.<span
style="mso-spacerun: yes"> </span>A-priori however must be the proof that if
(1) is solvable for some n such that n is greater than or equal to 3, then each
case of n may be “rewritten” (in possibly different<span style="mso-spacerun:
yes"> </span>r, a, b and n) as<span style="mso-spacerun: yes"> </span>v^n’ =
u^n’ + w^n’, where n’ is some divisor of n such that [v>u>w>0] is
necessary.<span style="mso-spacerun: yes"> </span>The different cases of n
will eventually be considered n’ such that n’ is greater than or equal to
3.<span style="mso-spacerun: yes"> </span></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify'><b>OBJECT</b>:
[(1) holds for any n such that n is greater than or equal to 3]<span
style="mso-spacerun: yes"> </span><u>implies</u><span style="mso-spacerun:
yes"> </span>(1) may be rewritten (in possibly different r, a, b and n)
as<span style="mso-spacerun: yes"> </span>v^n’ = u^n’ + w^n’,<span
style="mso-spacerun: yes"> </span>where n’ is greater than or equal to 3 such
that [v, u and w .GT. 0 ] is necessary.</p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='margin-left:15.0pt;text-align:justify;text-indent:
21.0pt'>CASE 1:<span style="mso-spacerun: yes"> </span>n has an odd divisor,
q, greater than 1.</p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify;text-indent:21.0pt'>Let
(1) be written, r^[(n/q)*q] = a^[(n/q)*q] + b^[(n/q)*q] and substituted as<span
style="mso-spacerun: yes"> </span>v^q = u^q + w^q.<span style="mso-spacerun:
yes"> </span>If n/q is even, the fact that<span style="mso-spacerun: yes">
</span>plus or minus r, plus or minus a and plus or minus b<span
style="mso-spacerun: yes"> </span>are roots of [v, u, w] implies the latter
are all positive.<span style="mso-spacerun: yes"> </span>On the other hand, if
n/q is odd, original conditions, [a>0 and b>0] imply the relation in
[v,u,w,n’] has strictly positive exponential bases.</p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify;text-indent:21.0pt'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify;text-indent:.5in'>CASE 2:<span
style="mso-spacerun: yes"> </span>n = 2^t where t is greater than or equal to
2.</p>
<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes"> </span><span style='mso-tab-count:1'> </span><span
style='mso-tab-count:1'> </span>Let(1) be written;</p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify;text-indent:1.75in'>[r^2^(t-1)]^
2 = [a^2^(t-1)]^2<span style="mso-spacerun: yes"> </span>+<span
style="mso-spacerun: yes"> </span>[b^2^(t-1)]^2<span style="mso-spacerun:
yes"> </span>which upon substitution becomes v^2 = u^2 + w^2.<span
style="mso-spacerun: yes"> </span>As before, the fact that 2^(t-1) is even and
the existence of roots plus or minus r, plus or minus a and plus or minus b,
corresponding to [v,u,w] imply they are all positive. Note that the case n=2
cannot be rewritten such that its root is not possibly negative.</p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify;text-indent:21.0pt'>Having
limited the existence of a positive-only solution for a rewrite of (1) such
that [r,a,b,n] goes to [v,u,w,n’]<span style="mso-spacerun: yes"> </span>for
all n greater than or equal to 3, order is specific and consequently
subject<span style="mso-spacerun: yes"> </span>to contradiction.</p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify'>Suppose now that
(1) has been rewritten for some n, where n is equal to or greater than n’ so
that [r,a,b and n] is transformed to a reduced form, redefining [r,a, b and n]
to be [u,v,w and n’]. Thus n’ is greater than or equal to 3 implies
[v>u>w>0].<span style="mso-spacerun: yes"> </span>Consider the
transformation (under n’), T, such that T:v to v, u to w, and w to u.
Respecting (1), T(v,u,w) = Tnot(v,u,w) where Tnot signifies not T.<span
style="mso-spacerun: yes"> </span>It follows that [u>w] can be transformed
to [w>u] such that [u>w] is unaltered.<b>!</b><span style="mso-spacerun:
yes"> </span>This can happen only for the case,u=w, which is moot.
Conclusively, order must be assignable when it exists. ( <b>!</b> denotes
contradiction.).</p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify;text-indent:21.0pt'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes">
</span></p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify'>SUPPOSE: [r is
conjugated negative in sign to [a]] which implies [(4) and (4)’ are indefinite]
allows a solution to (1) for the case n=2. </p>
<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes"> </span></p>
<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes">
</span>EXAMPLE<span style="mso-spacerun:
yes">
</span><span style="mso-spacerun:
yes"> </span></p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify'><span
style="mso-spacerun: yes"> </span>Let r^2 = a^2 + b^2<span
style="mso-spacerun: yes"> </span>(n=2).<span style="mso-spacerun: yes">
</span>Divide through by q^2, where q^2 is the greatest (square) divisor common
to [r^2,a^2 and b^2] to obtain the so-called “primitive” form, r^2/q ^2 = a^2/q
^2 + b^2/q ^2.<span style="mso-spacerun: yes"> </span>Rearrange as follows,
conjugating plus or minus |r| with plus or minus |a| where they are of
different sign respectively:</p>
<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes"> </span>|a|^2/q ^2 * [(-|r|)^2/|a|^2 – 1] =
(|a|^2)/q ^2)*[ |r|^2/a^2 – 1] = b^2/q ^2</p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify'><span
style="mso-spacerun: yes"> </span>b^2/q ^2 is substituted by 0, such that
a^2/q ^2, as an Integer multiple of 0, drops out. F is applied such that the
remaining is a relation among squares (preserving n).<span style="mso-spacerun:
yes"> </span>For example,</p>
<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes"> </span>F[(-5)^2/(4)^2 , 9] = F[4^2/4^2 , 9] =
F[5^2/(-4)^2 , 9] =</p>
<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes"> </span>F[(-4)^2/(-4)^2 , 9]</p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify'><span
style="mso-spacerun: yes"> </span>Consequently, F[1^2 , 9] – (-1)^2 =
0<span style="mso-spacerun: yes"> </span>is identifiable as a relation among
squares. Thus the presence of values altering the positivity of [r,a and b] allows
a solution to (1).<span style="mso-spacerun: yes"> </span></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify;text-indent:.5in'>In
the event that [v>u>w>0] (under n’) the latter method clearly cannot
succeed.<span style="mso-spacerun: yes"> </span>In any case, the instances
such that (1) is rewritten (for all non-trivial n, 3 or greater) stand
consistent while example accordingly excepts the case, n=2.</p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal align=center style='text-align:center'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal align=center style='text-align:center'>KEY</p>
<p class=MsoNormal align=center style='text-align:center'><b>!</b> denotes
contradiction</p>
<p class=MsoNormal align=center style='text-align:center'>|x| denotes the
absolute value of x</p>
<p class=MsoNormal align=center style='text-align:center'>* denotes
multiplication</p>
<p class=MsoNormal align=center style='text-align:center'>{x} denotes the
greatest Integer in x when x = |x|</p>
<p class=MsoNormal align=center style='text-align:center'>^ denotes
exponentiation</p>
<p class=MsoNormal align=center style='text-align:center'>Tnot denotes “not T”</p>
<p class=MsoNormal align=center style='text-align:center'>.LE. denotes “less
than or equal to”</p>
<p class=MsoNormal align=center style='text-align:center'>.GE. denotes “greater
than or equal to”</p>
<p class=MsoNormal align=center style='text-align:center'>.NE. denotes “not
equal to”</p>
<p class=MsoNormal><span style="mso-spacerun:
yes">
</span></p>
<p class=MsoNormal align=center style='text-align:center'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal align=center style='text-align:center'><span
style="mso-spacerun: yes"> </span></p>
<p class=MsoNormal align=center style='text-align:center'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal align=center style='text-align:center'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal align=center style='text-align:center'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'>-</p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='margin-left:.5in;text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes"> </span></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal style='text-align:justify'><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span></p>
<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal><span style="mso-spacerun: yes"> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style="mso-spacerun: yes"> </span></p>
<p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:1'> </span><span
style="mso-spacerun: yes"> </span><span style="mso-spacerun:
yes"> </span><span
style="mso-spacerun:
yes"> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:1'> </span><span
style='mso-tab-count:1'> </span>-<span style='mso-tab-count:1'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:2'> </span><span
style='mso-tab-count:2'> </span><span style='mso-tab-count:
2'> </span><span style='mso-tab-count:1'> </span></p>
<p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p class=MsoNormal><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
</div>
</body>
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