<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies ( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:122"/><span class="lab"><font color="Green" title="E98">A1</font></span>: 
<font color="Maroon" title="c3">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> 
 <span class="kw">and </span><br/><a NAME="E2:122"/><span class="lab"><font color="Green" title="E99">A2</font></span>: 
<span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<a NAME="E3:122"/><span class="lab"><font color="Green" title="E100">A3</font></span>: 
3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:122"><span class="lab"><font color="Green" title="E99">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:122"/><span class="lab"><font color="Green" title="E101">A4</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:122"><span class="lab"><font color="Green" title="E98">A1</font></span></a>, <a class="ref" href="abian.html#T9" onmouseover="rs('abian/T9')" onmouseout="rh()">ABIAN:9</a></span>;<br/>
<a NAME="E6:122"/>
3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">k</font></span>)</span>,1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T68" target="_self" onmouseover="rs('number08/T68')" onmouseout="rh()">Th68</a></span>;<br/>
<a NAME="E7:122"/><span class="kw">then </span>
<span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3,1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 3 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="number08.html#E13"><span class="lab"><font color="Green" title="E13">Lm13</font></span></a>, <a class="ref" href="int_1.html#T18" onmouseover="rs('int_1/T18')" onmouseout="rh()">INT_1:18</a></span>;<br/>
<a NAME="E8:122"/><span class="kw">then </span>
3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c3">n</font>,3 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:122"><span class="lab"><font color="Green" title="E101">A4</font></span></a>, <a class="ref" href="newton.html#T6" onmouseover="rs('newton/T6')" onmouseout="rh()">NEWTON:6</a></span>;<br/>
<a NAME="E9:122"/><span class="kw">then </span>
2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">m</font>,3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:122"><span class="lab"><font color="Green" title="E99">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 = 
2 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T64" onmouseover="rs('nat_d/T64')" onmouseout="rh()">NAT_D:64</a></span>
<br/>.= 
2
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</a></span>
;<br/>
<a NAME="E11:122"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A5</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T69" target="_self" onmouseover="rs('number08/T69')" onmouseout="rh()">Th69</a></span>;<br/>
<a NAME="E12:122"/><span class="kw">then </span>
3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:122"><span class="lab"><font color="Green" title="E100">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:122"/>
( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:122"><span class="lab"><font color="Green" title="E102">A5</font></span></a>, <a class="ref" href="pepin.html#T30" onmouseover="rs('pepin/T30')" onmouseout="rh()">PEPIN:30</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
