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<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:27_1_1_1"/>
<font color="Maroon" title="c1">X</font> is    <a href="pythtrip.html#M1" title="PYTHTRIP:mode.1">Pythagorean_triple</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</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="E3:27_1_1_1"/><i><font color="Green" title="E16">A1</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <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"><font color="Maroon" title="c3">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="c4">c</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> )
 <b>by </b><i><a class="ref" href="pythtrip.html#D4" target="_self" title="PYTHTRIP:def.4">Def4</a></i>;<br/><b>set </b><font color="Maroon" title="c5">k</font> = <font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font>;<br/><a NAME="E4:27_1_1_1"/><i><font color="Green" title="E17">A2</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1_1"/>
( <font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E18">A3</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="pythtrip.html#E4:27_1_1_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="int_2.html#T3" title="INT_2:th.3">INT_2:3</a></i>;<br/><b>thus </b><a NAME="E3:27_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">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>  st <br/>( <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">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="p4">(<span class="default"><font color="Olive" title="b2">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">n</font> <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="p4">(<span class="default"><font color="Olive" title="b2">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
;<br/>

<b>take </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
;<br/>

<b>take </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
;<br/>

<b>thus </b><a NAME="E1:27_1_1_1_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="pythtrip.html#E3:27_1_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="pythtrip.html#E2:27_1_1_1_1_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_1_2"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E19">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">a'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:27_1_1_1_1_2"/><i><font color="Green" title="E20">A6</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">a'</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E4:27_1_1_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">b'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:27_1_1_1_1_2"/><i><font color="Green" title="E21">A7</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">b'</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E4:27_1_1_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">a'</font> = <font color="Maroon" title="c6">a'</font>, <font color="Maroon" title="c9">b'</font> = <font color="Maroon" title="c7">b'</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/><a NAME="E8:27_1_1_1_1_2"/>
<font color="Maroon" title="c4">c</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a'</font> <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="p2">(<span class="default"><font color="Maroon" title="c9">b'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E3:27_1_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="pythtrip.html#E4:27_1_1_1_1_2"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="pythtrip.html#E6:27_1_1_1_1_2"><i><font color="Green" title="E21">A7</font></i></a></i>;<br/><a NAME="E9:27_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">c</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><a NAME="E10:27_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="ref" href="pythtrip.html#T6" target="_self" title="PYTHTRIP:th.6">Th6</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">c'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E12:27_1_1_1_1_2"/><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">c'</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">c'</font> = <font color="Maroon" title="c10">c'</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/><a NAME="E14:27_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a'</font> <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="p2">(<span class="default"><font color="Maroon" title="c9">b'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">c'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E3:27_1_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="pythtrip.html#E4:27_1_1_1_1_2"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="pythtrip.html#E6:27_1_1_1_1_2"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="pythtrip.html#E12:27_1_1_1_1_2"><i><font color="Green" title="E22">A8</font></i></a></i>;<br/><a NAME="E15:27_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E23">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a'</font> <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"><font color="Maroon" title="c9">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="c11">c'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:27_1_1_1_1_2"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E16:27_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a'</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c9">b'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1
 <b>by </b><i><a class="txt" href="pythtrip.html#E4:27_1_1_1_1_2"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="pythtrip.html#E6:27_1_1_1_1_2"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="pythtrip.html#T8" target="_self" title="PYTHTRIP:th.8">Th8</a></i>;<br/><a NAME="E17:27_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">a'</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c9">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="pythtrip.html#E1:27_1_1_1_1_2"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E18:27_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E24">A10</font></i>: 
( <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  &amp; <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c8">a'</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )
 <b>by </b><i><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></i>;<br/><b>thus </b><a NAME="E19:27_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">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>  st <br/>( <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">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="p4">(<span class="default"><font color="Olive" title="b2">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">n</font> <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="p4">(<span class="default"><font color="Olive" title="b2">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1_1_2_1_1"/>
(  not <font color="Maroon" title="c8">a'</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or  not <font color="Maroon" title="c9">b'</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 </span><b>by </b><i><a class="txt" href="pythtrip.html#E18:27_1_1_1_1_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="pythtrip.html#T10" target="_self" title="PYTHTRIP:th.10">Th10</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_1_2_1_1_1"/>
not <font color="Maroon" title="c8">a'</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">m</font>, <font color="Maroon" title="c13">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="E3:27_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E25">A11</font></i>: 
( <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c8">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c12">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> &amp; <font color="Maroon" title="c9">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c11">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <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"><font color="Maroon" title="c12">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#E15:27_1_1_1_1_2"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="pythtrip.html#E18:27_1_1_1_1_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="pythtrip.html#T11" target="_self" title="PYTHTRIP:th.11">Th11</a></i>;<br/><b>take </b>
<font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font>
;<br/><b>take </b>
<font color="Maroon" title="c12">m</font>
;<br/><b>take </b>
<font color="Maroon" title="c13">n</font>
;<br/><b>thus </b><a NAME="E4:27_1_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">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="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">n</font> <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="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="pythtrip.html#E3:27_1_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="pythtrip.html#E4:27_1_1_1_1_2"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="pythtrip.html#E6:27_1_1_1_1_2"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="pythtrip.html#E12:27_1_1_1_1_2"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="pythtrip.html#E3:27_1_1_1_1_2_1_1_1"><i><font color="Green" title="E25">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_1_2_1_1_2"/>
not <font color="Maroon" title="c9">b'</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">m</font>, <font color="Maroon" title="c13">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="E3:27_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E25">A12</font></i>: 
( <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c9">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c12">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> &amp; <font color="Maroon" title="c8">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c11">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <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"><font color="Maroon" title="c12">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#E15:27_1_1_1_1_2"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="pythtrip.html#E18:27_1_1_1_1_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="pythtrip.html#T11" target="_self" title="PYTHTRIP:th.11">Th11</a></i>;<br/><b>take </b>
<font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font>
;<br/><b>take </b>
<font color="Maroon" title="c12">m</font>
;<br/><b>take </b>
<font color="Maroon" title="c13">n</font>
;<br/><b>thus </b><a NAME="E4:27_1_1_1_1_2_1_1_2"/>
( <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">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="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">n</font> <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="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="pythtrip.html#E3:27_1_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="pythtrip.html#E4:27_1_1_1_1_2"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="pythtrip.html#E6:27_1_1_1_1_2"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="pythtrip.html#E12:27_1_1_1_1_2"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="pythtrip.html#E3:27_1_1_1_1_2_1_1_2"><i><font color="Green" title="E25">A12</font></i></a>, <a class="ref" href="enumset1.html#T99" title="ENUMSET1:th.99">ENUMSET1:99</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:27_1_1"/>
 ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">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>  st <br/>( <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">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="p4">(<span class="default"><font color="Olive" title="b2">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">n</font> <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="p4">(<span class="default"><font color="Olive" title="b2">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">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="E4:27_1_1"/><i><font color="Green" title="E16">A13</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">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="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <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="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> )
 ;<br/>
<a NAME="E5:27_1_1"/>
<font color="Maroon" title="c3">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="c4">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 
<b>by </b><i><a class="txt" href="pythtrip.html#E4:27_1_1"><i><font color="Green" title="E16">A13</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">a'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">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="c3">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> 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>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">b'</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">n</font>;<br/>
<b>set </b><font color="Maroon" title="c7">c'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <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"><font color="Maroon" title="c3">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">a'</font>;<br/>
<b>set </b><font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <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="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span>;<br/>
<a NAME="E7:27_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">a'</font></span>)</span> <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="c2">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <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="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 
;<br/>
<b>hence </b><a NAME="E8:27_1_1"/>
<font color="Maroon" title="c1">X</font> is    <a href="pythtrip.html#M1" title="PYTHTRIP:mode.1">Pythagorean_triple</a> 
 <b>by </b><i><a class="txt" href="pythtrip.html#E4:27_1_1"><i><font color="Green" title="E16">A13</font></i></a>, <a class="ref" href="pythtrip.html#D4" target="_self" title="PYTHTRIP:def.4">Def4</a></i>;<br/>


</div>
