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<div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:27_1_1_1"/>
<font color="Maroon">X</font> is    <a href="pythtrip.html#M1" target="_self">Pythagorean_triple</a> 
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:27_1_1_1"/><i><font color="Green">E30</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="pepin.html#K2">^2</a>  &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K8">}</a></span> )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">k</font> = <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>;<br/><a NAME="E4:27_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1: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">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </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">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="int_2.html#T3">INT_2:3</a></i>;<br/><b>thus </b><a NAME="E3:27_1_1_1_1_1"/>
 ex <font color="Olive">k</font>, <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Olive">m</font> <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Olive">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Olive">m</font> <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="enumset1.html#K1">}</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>
0
;<br/>

<b>take </b>
0
;<br/>

<b>take </b>
0
;<br/>

<b>thus </b><a NAME="E1:27_1_1_1_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">0 <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">0 <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default">0 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K2">*</a> 0</span>)</span></span>)</span>,<span class="p2">(<span class="default">0 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">0 <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default">0 <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="enumset1.html#K1">}</a></span> )
 <b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T6">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">E34</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><a NAME="E3:27_1_1_1_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="pepin.html#K2">^2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><b>consider </b><font color="Maroon">a'</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:27_1_1_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a'</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">b'</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:27_1_1_1_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b'</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a'</font>, <font color="Maroon">b'</font> = <font color="Maroon">b'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E9:27_1_1_1_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c</font> <a href="pepin.html#K2">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E10:27_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="pepin.html#K2">^2</a>  <a href="nat_1.html#R1">divides</a> <font color="Maroon">c</font> <a href="pepin.html#K2">^2</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E11:27_1_1_1_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">c'</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:27_1_1_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c'</font> = <font color="Maroon">c'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E15:27_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c'</font> <a href="pepin.html#K2">^2</a> </span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E16:27_1_1_1_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font> <a href="pepin.html#K2">^2</a> 
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E17:27_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E18:27_1_1_1_1_2"/><b>then </b>
<font color="Maroon">a'</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E19:27_1_1_1_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Maroon">b'</font>,<font color="Maroon">a'</font> <a href="int_2.html#R2">are_relative_prime</a>  )
 <b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/><b>thus </b><a NAME="E20:27_1_1_1_1_2"/>
 ex <font color="Olive">k</font>, <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Olive">m</font> <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Olive">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Olive">m</font> <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="enumset1.html#K1">}</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">a'</font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">b'</font> is <a href="abian.html#V1">even</a> )
 </span><b>by </b><i>, </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">a'</font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:27_1_1_1_1_2_1_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span> )
 <b>by </b><i>, , </i>;<br/><b>take </b>
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>
;<br/><b>take </b>
<font color="Maroon">m</font>
;<br/><b>take </b>
<font color="Maroon">n</font>
;<br/><b>thus </b><a NAME="E4:27_1_1_1_1_2_1_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="enumset1.html#K1">}</a></span> )
 <b>by </b><i>, , , , </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">b'</font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:27_1_1_1_1_2_1_1_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span> )
 <b>by </b><i>, , </i>;<br/><b>take </b>
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>
;<br/><b>take </b>
<font color="Maroon">m</font>
;<br/><b>take </b>
<font color="Maroon">n</font>
;<br/><b>thus </b><a NAME="E4:27_1_1_1_1_2_1_1_2"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="enumset1.html#K1">}</a></span> )
 <b>by </b><i>, , , , , <a class="ref" href="enumset1.html#T99">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">k</font>, <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Olive">m</font> <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Olive">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Olive">m</font> <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="enumset1.html#K1">}</a></span> )
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:27_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="enumset1.html#K1">}</a></span> )
 ;<br/>
<a NAME="E5:27_1_1"/>
<font color="Maroon">m</font> <a href="pepin.html#K2">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> 
 
<b>by </b><i>, <a class="ref" href="square_1.html#T77">SQUARE_1:77</a></i>;<br/>
<a NAME="E6:27_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a'</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<b>set </b><font color="Maroon">b'</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">c'</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">a</font> = <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font>;<br/>
<b>set </b><font color="Maroon">b</font> = <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c</font> = <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span>;<br/>
<a NAME="E8:27_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span> <a href="pepin.html#K2">^2</a> 
 
;<br/>
<b>hence </b><a NAME="E9:27_1_1"/>
<font color="Maroon">X</font> is    <a href="pythtrip.html#M1" target="_self">Pythagorean_triple</a> 
 <b>by </b><i>, </i>;<br/>


</div>
