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

<b>set </b><font color="Maroon">T</font> = <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is    <a href="pythtrip.html#M1" target="_self">Pythagorean_triple</a>  : ( not <font color="Olive">X</font> is <a href="pythtrip.html#V2" target="_self">degenerate</a> &amp; <font color="Olive">X</font> is <a href="pythtrip.html#V3" target="_self">simplified</a> ) </span>}</span> ;<br/>
<a NAME="E1:35"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <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">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is    <a href="pythtrip.html#M1" target="_self">Pythagorean_triple</a>  : ( not <font color="Olive">X</font> is <a href="pythtrip.html#V2" target="_self">degenerate</a> &amp; <font color="Olive">X</font> is <a href="pythtrip.html#V3" target="_self">simplified</a> ) </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>set </b><font color="Maroon">m'</font> = <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="pythtrip.html#M1" target="_self">Pythagorean_triple</a> <b> such that </b><br/><a NAME="E2:35_1"/><i><font color="Green">E30</font></i>: 
( not <font color="Maroon">X</font> is <a href="pythtrip.html#V2" target="_self">degenerate</a> &amp; <font color="Maroon">X</font> is <a href="pythtrip.html#V3" target="_self">simplified</a> &amp; 4 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">n</font> = 4 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>take </b>
4 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
;<br/>

<a NAME="E3:35_1"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
;<br/>
<a NAME="E4:35_1"/><i><font color="Green">E33</font></i>: 
4 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:35_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E6:35_1"/>
4 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>

<a NAME="E7:35_1"/>
( 4 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is    <a href="pythtrip.html#M1" target="_self">Pythagorean_triple</a>  : ( not <font color="Olive">X</font> is <a href="pythtrip.html#V2" target="_self">degenerate</a> &amp; <font color="Olive">X</font> is <a href="pythtrip.html#V3" target="_self">simplified</a> ) </span>}</span>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:35_1"/>
4 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is    <a href="pythtrip.html#M1" target="_self">Pythagorean_triple</a>  : ( not <font color="Olive">X</font> is <a href="pythtrip.html#V2" target="_self">degenerate</a> &amp; <font color="Olive">X</font> is <a href="pythtrip.html#V3" target="_self">simplified</a> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:35"/><b>then </b><i><font color="Green">E34</font></i>: 
not  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is    <a href="pythtrip.html#M1" target="_self">Pythagorean_triple</a>  : ( not <font color="Olive">X</font> is <a href="pythtrip.html#V2" target="_self">degenerate</a> &amp; <font color="Olive">X</font> is <a href="pythtrip.html#V3" target="_self">simplified</a> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:35_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is    <a href="pythtrip.html#M1" target="_self">Pythagorean_triple</a>  : ( not <font color="Olive">X</font> is <a href="pythtrip.html#V2" target="_self">degenerate</a> &amp; <font color="Olive">X</font> is <a href="pythtrip.html#V3" target="_self">simplified</a> ) </span>}</span> 
 ;<br/><a NAME="E2:35_2"/><b>then </b>
 ex <font color="Olive">X'</font> being   <a href="pythtrip.html#M1" target="_self">Pythagorean_triple</a>  st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Olive">X'</font> &amp; not <font color="Olive">X'</font> is <a href="pythtrip.html#V2" target="_self">degenerate</a> &amp; <font color="Olive">X'</font> is <a href="pythtrip.html#V3" target="_self">simplified</a> )
 ;<br/><b>hence </b><a NAME="E3:35_2"/>
<font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:35"/>
not <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is    <a href="pythtrip.html#M1" target="_self">Pythagorean_triple</a>  : ( not <font color="Olive">X</font> is <a href="pythtrip.html#V2" target="_self">degenerate</a> &amp; <font color="Olive">X</font> is <a href="pythtrip.html#V3" target="_self">simplified</a> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/>


</div>
