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<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">X</font> is <a href="pythtrip.html#V3" target="_self">simplified</a>
 ;<br/><b>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="E3:31_1_1_1"/><i><font color="Green">E31</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>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>take </b><font color="Maroon">b</font> = <font color="Maroon">b</font>;<br/><b>thus </b><a NAME="E4:31_1_1_1"/>
<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/><b>thus </b><a NAME="E5:31_1_1_1"/>
<font color="Maroon">b</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:31_1_1_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font> )
 ;<br/><b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k</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="E3:31_1_1_1_1"/>
( <font color="Maroon">k1</font> <a href="pepin.html#K2">^2</a>  <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> <a href="pepin.html#K2">^2</a>  &amp; <font color="Maroon">k1</font> <a href="pepin.html#K2">^2</a>  <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font> <a href="pepin.html#K2">^2</a>  )
 <b>by </b><i>, </i>;<br/><a NAME="E4:31_1_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <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#T55">NAT_1:55</a></i>;<br/><a NAME="E5:31_1_1_1_1"/><b>then </b>
<font color="Maroon">k1</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/><a NAME="E6:31_1_1_1_1"/><b>then </b>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">k1</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">n</font>
 <b>by </b><i>, , <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><b>hence </b><a NAME="E7:31_1_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:31_1_1_1"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:31_1_1"/>
 ex <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="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">m</font>,<font color="Olive">n</font> <a href="int_2.html#R2">are_relative_prime</a>  )
 ;<br/>

<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="E4:31_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="int_2.html#R2">are_relative_prime</a>  )
 ;<br/>
<a NAME="E5:31_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pythtrip.html#D7" target="_self">PYTHTRIP:def 7</a><br/>

<b>assume </b><a NAME="E6:31_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">n</font>
 ;<br/>

<a NAME="E7:31_1_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E8:31_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:31_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="wsierp_1.html#T20">WSIERP_1:20</a></i>;<br/>


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