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

<b>set </b><font color="Maroon" title="c1">T</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is    <a href="pythtrip.html#M1" title="PYTHTRIP:mode.1">Pythagorean_triple</a>  : ( not <font color="Olive" title="b1">X</font> is <a href="pythtrip.html#V2" title="PYTHTRIP:attr.2">degenerate</a> &amp; <font color="Olive" title="b1">X</font> is <a href="pythtrip.html#V3" title="PYTHTRIP:attr.3">simplified</a> ) </span>}</span> ;<br/>
<a NAME="E1:35"/>
for <font color="Olive" title="b1">m</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>   ex <font color="Olive" title="b2">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">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">X</font> where <font color="Olive" title="b3">X</font> is    <a href="pythtrip.html#M1" title="PYTHTRIP:mode.1">Pythagorean_triple</a>  : ( not <font color="Olive" title="b3">X</font> is <a href="pythtrip.html#V2" title="PYTHTRIP:attr.2">degenerate</a> &amp; <font color="Olive" title="b3">X</font> is <a href="pythtrip.html#V3" title="PYTHTRIP:attr.3">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" title="c2">m</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">X</font> where <font color="Olive" title="b2">X</font> is    <a href="pythtrip.html#M1" title="PYTHTRIP:mode.1">Pythagorean_triple</a>  : ( not <font color="Olive" title="b2">X</font> is <a href="pythtrip.html#V2" title="PYTHTRIP:attr.2">degenerate</a> &amp; <font color="Olive" title="b2">X</font> is <a href="pythtrip.html#V3" title="PYTHTRIP:attr.3">simplified</a> ) </span>}</span>  )</span><br/>

<b>set </b><font color="Maroon" title="c3">m'</font> = <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<b>consider </b><font color="Maroon" title="c4">X</font> being    <a href="pythtrip.html#M1" title="PYTHTRIP:mode.1">Pythagorean_triple</a> <b> such that </b><br/><a NAME="E2:35_1"/><i><font color="Green" title="E21">A1</font></i>: 
( not <font color="Maroon" title="c4">X</font> is <a href="pythtrip.html#V2" title="PYTHTRIP:attr.2">degenerate</a> &amp; <font color="Maroon" title="c4">X</font> is <a href="pythtrip.html#V3" title="PYTHTRIP:attr.3">simplified</a> &amp; 4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )
 <b>by </b><i><a class="ref" href="pythtrip.html#T14" target="_self" title="PYTHTRIP:th.14">Th14</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">n</font> = 4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<b>take </b>
4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">m</font> &amp; 4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is    <a href="pythtrip.html#M1" title="PYTHTRIP:mode.1">Pythagorean_triple</a>  : ( not <font color="Olive" title="b1">X</font> is <a href="pythtrip.html#V2" title="PYTHTRIP:attr.2">degenerate</a> &amp; <font color="Olive" title="b1">X</font> is <a href="pythtrip.html#V3" title="PYTHTRIP:attr.3">simplified</a> ) </span>}</span>  )</span><br/>

<a NAME="E3:35_1"/>
<span class="p1">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
;<br/>
<a NAME="E4:35_1"/><b>then </b><i><font color="Green" title="E22">A2</font></i>: 
4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:35_1"/>
<font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<b>hence </b><a NAME="E6:35_1"/>
4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E4:35_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is    <a href="pythtrip.html#M1" title="PYTHTRIP:mode.1">Pythagorean_triple</a>  : ( not <font color="Olive" title="b1">X</font> is <a href="pythtrip.html#V2" title="PYTHTRIP:attr.2">degenerate</a> &amp; <font color="Olive" title="b1">X</font> is <a href="pythtrip.html#V3" title="PYTHTRIP:attr.3">simplified</a> ) </span>}</span> </span><br/>

<a NAME="E7:35_1"/>
( 4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is    <a href="pythtrip.html#M1" title="PYTHTRIP:mode.1">Pythagorean_triple</a>  : ( not <font color="Olive" title="b1">X</font> is <a href="pythtrip.html#V2" title="PYTHTRIP:attr.2">degenerate</a> &amp; <font color="Olive" title="b1">X</font> is <a href="pythtrip.html#V3" title="PYTHTRIP:attr.3">simplified</a> ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="pythtrip.html#E2:35_1"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:35_1"/>
4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is    <a href="pythtrip.html#M1" title="PYTHTRIP:mode.1">Pythagorean_triple</a>  : ( not <font color="Olive" title="b1">X</font> is <a href="pythtrip.html#V2" title="PYTHTRIP:attr.2">degenerate</a> &amp; <font color="Olive" title="b1">X</font> is <a href="pythtrip.html#V3" title="PYTHTRIP:attr.3">simplified</a> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:35"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
not  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is    <a href="pythtrip.html#M1" title="PYTHTRIP:mode.1">Pythagorean_triple</a>  : ( not <font color="Olive" title="b1">X</font> is <a href="pythtrip.html#V2" title="PYTHTRIP:attr.2">degenerate</a> &amp; <font color="Olive" title="b1">X</font> is <a href="pythtrip.html#V3" title="PYTHTRIP:attr.3">simplified</a> ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="pythtrip.html#T9" target="_self" title="PYTHTRIP:th.9">Th9</a></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" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is    <a href="pythtrip.html#M1" title="PYTHTRIP:mode.1">Pythagorean_triple</a>  : ( not <font color="Olive" title="b1">X</font> is <a href="pythtrip.html#V2" title="PYTHTRIP:attr.2">degenerate</a> &amp; <font color="Olive" title="b1">X</font> is <a href="pythtrip.html#V3" title="PYTHTRIP:attr.3">simplified</a> ) </span>}</span>  implies <font color="Maroon" title="c2">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/><b>assume </b><a NAME="E1:35_2"/>
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is    <a href="pythtrip.html#M1" title="PYTHTRIP:mode.1">Pythagorean_triple</a>  : ( not <font color="Olive" title="b1">X</font> is <a href="pythtrip.html#V2" title="PYTHTRIP:attr.2">degenerate</a> &amp; <font color="Olive" title="b1">X</font> is <a href="pythtrip.html#V3" title="PYTHTRIP:attr.3">simplified</a> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/><a NAME="E2:35_2"/><b>then </b>
 ex <font color="Olive" title="b1">X'</font> being   <a href="pythtrip.html#M1" title="PYTHTRIP:mode.1">Pythagorean_triple</a>  st <br/>( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X'</font> &amp; not <font color="Olive" title="b1">X'</font> is <a href="pythtrip.html#V2" title="PYTHTRIP:attr.2">degenerate</a> &amp; <font color="Olive" title="b1">X'</font> is <a href="pythtrip.html#V3" title="PYTHTRIP:attr.3">simplified</a> )
 ;<br/><b>hence </b><a NAME="E3:35_2"/>
<font color="Maroon" title="c2">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:35"/>
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is    <a href="pythtrip.html#M1" title="PYTHTRIP:mode.1">Pythagorean_triple</a>  : ( not <font color="Olive" title="b1">X</font> is <a href="pythtrip.html#V2" title="PYTHTRIP:attr.2">degenerate</a> &amp; <font color="Olive" title="b1">X</font> is <a href="pythtrip.html#V3" title="PYTHTRIP:attr.3">simplified</a> ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:35"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="finset_1.html#T25" title="FINSET_1:th.25">FINSET_1:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
