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<span class="p1">(<span class="default">3 <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">4 <a href="pepin.html#K2">^2</a> </span>)</span> = 
<span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> 3</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">4 <a href="pepin.html#K2">^2</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> 3</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> 4</span>)</span>

<br/>.= 
5 <a href="nat_1.html#K2">*</a> 5

<br/>.= 
5 <a href="pepin.html#K2">^2</a> 

;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">3,4,5</span><a href="domain_1.html#K8">}</a></span> as    <a href="pythtrip.html#M1" target="_self">Pythagorean_triple</a>  <b>by </b><i><a class="ref" href="pythtrip.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:34"/><i><font color="Green">E21</font></i>: 
not 0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
3 <a href="nat_1.html#K8">hcf</a> 4 = 
<span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> 3</span>)</span> <a href="nat_1.html#K8">hcf</a> 4
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> 3</span>)</span> <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> 4</span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
3 <a href="int_2.html#K3">gcd</a> 4

<br/>.= 
3 <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K5">-</a> 3</span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T111">PREPOWER:111</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="wsierp_1.html#T13">WSIERP_1:13</a></i>
;<br/>
<a NAME="E5:34"/><b>then </b>
( 3 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; 4 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; 3,4 <a href="int_2.html#R2">are_relative_prime</a>  )
 
<b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E6:34"/>
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default">3,4,5</span><a href="domain_1.html#K8">}</a></span> is  non <a href="pythtrip.html#V2" target="_self">degenerate</a> <a href="pythtrip.html#V3" target="_self">simplified</a>  <a href="pythtrip.html#M1" target="_self">Pythagorean_triple</a> 
 <b>by </b><i><a class="txt" href="pythtrip.html#E3:34"><i><font color="Green">E21</font></i></a>, <a class="ref" href="pythtrip.html#D6" target="_self">Def6</a>, <a class="ref" href="pythtrip.html#D8" target="_self">Def8</a></i>;<br/>


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