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

<b>assume </b><a NAME="E1:32"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:32"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:32"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E4:32"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="pepin.html#K2">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="pepin.html#K2">^2</a> 
 
<b>by </b><i><a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/>
<a NAME="E5:32"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = 1 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">1 <a href="pepin.html#K2">^2</a> </span>)</span></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> = 1 <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> 1</span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c</font> = 1 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default">1 <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> 1</span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default">1 <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K8">}</a></span> as    <a href="pythtrip.html#M1" target="_self">Pythagorean_triple</a>  <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<a NAME="E8:32"/><i><font color="Green">E34</font></i>: 
1 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <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">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E9:32"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E10:32"/>
1 <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> 1</span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:32"/>
not 0 <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pythtrip.html#D6" target="_self">PYTHTRIP:def 6</a><br/>

<a NAME="E12:32"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E13:32"/><i><font color="Green">E42</font></i>: 
1 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E14:32"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> 2
 
;<br/>
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="prepower.html#T111">PREPOWER:111</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="int_2.html#K3">gcd</a> 2
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1 <a href="int_2.html#K3">gcd</a> 2
<b>by </b><i><a class="ref" href="euler_1.html#T17">EULER_1:17</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="wsierp_1.html#T13">WSIERP_1:13</a></i>
;<br/>
<a NAME="E16:32"/><b>then </b>
<font color="Maroon">a</font>,1 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="pepin.html#K2">^2</a> </span>)</span></span>)</span> <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>hence </b><a NAME="E17:32"/>
<font color="Maroon">X</font> is <a href="pythtrip.html#V3" target="_self">simplified</a>
 <b>by </b><i>, , </i>;<br/>

<b>thus </b><a NAME="E18:32"/>
4 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


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