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

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E15">A1</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 ;<br/>

<b>assume </b><a NAME="E2:24"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 ;<br/>

<b>assume </b><a NAME="E3:24"/>
not <font color="Maroon" title="c1">a</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">a'</font> = <font color="Maroon" title="c1">a</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E5:24"/>
<font color="Maroon" title="c2">b</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1"/>
not <font color="Maroon" title="c2">b</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">b'</font> = <font color="Maroon" title="c2">b</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:24_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 
<b>by </b><i><a class="txt" href="pythtrip.html#E1:24"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:24_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">b'</font> = <font color="Maroon" title="c2">b</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E7:24"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 
<b>by </b><i><a class="txt" href="pythtrip.html#E1:24"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">c'</font> = <font color="Maroon" title="c3">c</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="pythtrip.html#T2" target="_self" title="PYTHTRIP:th.2">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">n'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:24"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c6">c'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">n'</font>
 <b>by </b><i><a class="ref" href="abian.html#D2" title="ABIAN:def.2">ABIAN:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E12:24"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c6">c'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="ref" href="abian.html#D1" title="ABIAN:def.1">ABIAN:def 1</a></i>;<br/>
<a NAME="E13:24"/>
<font color="Maroon" title="c3">c</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
 
<b>by </b><i><a class="txt" href="pythtrip.html#E1:24"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E14:24"/><b>then </b>
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="ref" href="square_1.html#T78" title="SQUARE_1:th.78">SQUARE_1:78</a></i>;<br/>
<a NAME="E15:24"/><b>then </b>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0
 
<b>by </b><i><a class="txt" href="pythtrip.html#E12:24"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E16:24"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">m'</font> = <font color="Maroon" title="c8">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">k'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E19:24"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c5">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c10">k'</font>
 <b>by </b><i><a class="ref" href="abian.html#D2" title="ABIAN:def.2">ABIAN:def 2</a></i>;<br/>
<a NAME="E20:24"/><i><font color="Green" title="E20">A6</font></i>: 
<font color="Maroon" title="c7">n'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">m'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="pythtrip.html#E10:24"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="pythtrip.html#E12:24"><i><font color="Green" title="E18">A4</font></i></a></i>;<br/>
<a NAME="E21:24"/><i><font color="Green" title="E21">A7</font></i>: 
<font color="Maroon" title="c7">n'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">m'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 
<b>by </b><i><a class="txt" href="pythtrip.html#E10:24"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="pythtrip.html#E12:24"><i><font color="Green" title="E18">A4</font></i></a></i>;<br/>
<a NAME="E22:24"/><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c7">n'</font>,<font color="Maroon" title="c9">m'</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">p</font> be  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="pythtrip.html#D2" target="_self" title="PYTHTRIP:def.2">PYTHTRIP:def 2</a><br/>

<b>assume </b><a NAME="E1:24_2"/><i><font color="Green" title="E23">A9</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">n'</font> &amp; <font color="Maroon" title="c11">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c9">m'</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c12">p</font> = <font color="Maroon" title="c11">p</font> as  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:24_2"/>
<font color="Maroon" title="c12">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c9">m'</font>
 
<b>by </b><i><a class="txt" href="pythtrip.html#E1:24_2"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="int_2.html#T14" title="INT_2:th.14">INT_2:14</a></i>;<br/>
<a NAME="E4:24_2"/><b>then </b><i><font color="Green" title="E24">A10</font></i>: 
<font color="Maroon" title="c12">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c7">n'</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c9">m'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pythtrip.html#E1:24_2"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="wsierp_1.html#T9" title="WSIERP_1:th.9">WSIERP_1:9</a></i>;<br/>
<a NAME="E5:24_2"/><b>then </b>
<font color="Maroon" title="c12">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="pythtrip.html#E10:24"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="pythtrip.html#E12:24"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>;<br/>
<a NAME="E6:24_2"/><b>then </b><i><font color="Green" title="E25">A11</font></i>: 
<font color="Maroon" title="c12">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_2.html#T14" title="INT_2:th.14">INT_2:14</a></i>;<br/>
<a NAME="E7:24_2"/>
<font color="Maroon" title="c12">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">c</font>
 
<b>by </b><i><a class="txt" href="pythtrip.html#E21:24"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="pythtrip.html#E1:24_2"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="nat_d.html#T8" title="NAT_D:th.8">NAT_D:8</a></i>;<br/>
<a NAME="E8:24_2"/><b>then </b><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c12">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">c</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>;<br/>
<a NAME="E9:24_2"/>
<font color="Maroon" title="c2">b</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pythtrip.html#E1:24"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/>
<a NAME="E10:24_2"/><b>then </b>
<font color="Maroon" title="c12">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="txt" href="pythtrip.html#E6:24_2"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="pythtrip.html#E8:24_2"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="wsierp_1.html#T9" title="WSIERP_1:th.9">WSIERP_1:9</a></i>;<br/>
<a NAME="E11:24_2"/><b>then </b>
<font color="Maroon" title="c12">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="newton.html#T98" title="NEWTON:th.98">NEWTON:98</a></i>;<br/>
<b>hence </b><a NAME="E12:24_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:24"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="pythtrip.html#E10:24"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="pythtrip.html#E12:24"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="pythtrip.html#E4:24_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="pythtrip.html#D2" target="_self" title="PYTHTRIP:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green" title="E23">A13</font></i>: <font color="Maroon" title="c7">n'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c9">m'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="pythtrip.html#E10:24"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="pythtrip.html#E12:24"><i><font color="Green" title="E18">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
<b>by </b><i><a class="txt" href="pythtrip.html#E1:24"><i><font color="Green" title="E15">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">k'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
<b>by </b><i><a class="txt" href="pythtrip.html#E19:24"><i><font color="Green" title="E19">A5</font></i></a></i>
;<br/>
<a NAME="E24:24"/><b>then </b><i><font color="Green" title="E24">A14</font></i>: 
( <font color="Maroon" title="c7">n'</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> &amp; <font color="Maroon" title="c9">m'</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> )
 
<b>by </b><i><a class="txt" href="pythtrip.html#E22:24"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="pythtrip.html#T1" target="_self" title="PYTHTRIP:th.1">Th1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E26:24"/><i><font color="Green" title="E25">A15</font></i>: 
<font color="Maroon" title="c7">n'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <b>by </b><i><a class="ref" href="pythtrip.html#D3" target="_self" title="PYTHTRIP:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E28:24"/><i><font color="Green" title="E26">A16</font></i>: 
<font color="Maroon" title="c9">m'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <b>by </b><i><a class="txt" href="pythtrip.html#E24:24"><i><font color="Green" title="E24">A14</font></i></a>, <a class="ref" href="pythtrip.html#D3" target="_self" title="PYTHTRIP:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">m</font> = <font color="Maroon" title="c12">m</font>, <font color="Maroon" title="c14">n</font> = <font color="Maroon" title="c11">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">m</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c14">n</font>
;<br/>

<a NAME="E30:24"/>
( <font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c13">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  )
 
<b>by </b><i><a class="txt" href="pythtrip.html#E20:24"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="pythtrip.html#E26:24"><i><font color="Green" title="E25">A15</font></i></a>, <a class="txt" href="pythtrip.html#E28:24"><i><font color="Green" title="E26">A16</font></i></a>, <a class="ref" href="real_2.html#T105" title="REAL_2:th.105">REAL_2:105</a></i>;<br/>
<b>hence </b><a NAME="E31:24"/>
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="ref" href="square_1.html#T78" title="SQUARE_1:th.78">SQUARE_1:78</a></i>;<br/>

<b>thus </b><a NAME="E32:24"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E10:24"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="pythtrip.html#E12:24"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="pythtrip.html#E26:24"><i><font color="Green" title="E25">A15</font></i></a>, <a class="txt" href="pythtrip.html#E28:24"><i><font color="Green" title="E26">A16</font></i></a></i>;<br/>

<font color="Maroon" title="c2">b</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  = 
<span class="p1">(<span class="default">2 <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="pythtrip.html#E19:24"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="pythtrip.html#E23:24"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="square_1.html#T68" title="SQUARE_1:th.68">SQUARE_1:68</a>, <a class="txt" href="pythtrip.html#E26:24"><i><font color="Green" title="E25">A15</font></i></a>, <a class="txt" href="pythtrip.html#E28:24"><i><font color="Green" title="E26">A16</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 

;<br/>
<b>hence </b><a NAME="E34:24"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="ref" href="pythtrip.html#T5" target="_self" title="PYTHTRIP:th.5">Th5</a></i>;<br/>

<b>thus </b><a NAME="E35:24"/>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E10:24"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="pythtrip.html#E12:24"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="pythtrip.html#E26:24"><i><font color="Green" title="E25">A15</font></i></a>, <a class="txt" href="pythtrip.html#E28:24"><i><font color="Green" title="E26">A16</font></i></a></i>;<br/>


</div>
