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<div><b>hereby </b> <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">m</font>, <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="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  ) implies <font color="Maroon" title="c1">X</font> is <a href="pythtrip.html#V3" title="PYTHTRIP:attr.3">simplified</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:30_1_1_1"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="pythtrip.html#V3" title="PYTHTRIP:attr.3">simplified</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">a</font>, <font color="Olive" title="b2">b</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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )</span><br/><b>consider </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</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> <b> such that </b><br/><a NAME="E3:30_1_1_1"/><i><font color="Green" title="E19">A2</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> )
 <b>by </b><i><a class="ref" href="pythtrip.html#D4" target="_self" title="PYTHTRIP:def.4">Def4</a></i>;<br/><b>take </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c2">a</font>; <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">b</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="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )</span><br/><b>take </b><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c3">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )</span><br/><b>thus </b><a NAME="E4:30_1_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E3:30_1_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )</span><br/><b>thus </b><a NAME="E5:30_1_1_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E3:30_1_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c7">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><b>assume </b><a NAME="E1:30_1_1_1_1"/><i><font color="Green" title="E20">A3</font></i>: 
( <font color="Maroon" title="c7">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c7">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><b>reconsider </b><font color="Maroon" title="c8">k1</font> = <font color="Maroon" title="c7">k</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:30_1_1_1_1"/>
( <font color="Maroon" title="c8">k1</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">a</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  &amp; <font color="Maroon" title="c8">k1</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">b</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  )
 <b>by </b><i><a class="txt" href="#E1:30_1_1_1_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="pythtrip.html#T6" target="_self" title="PYTHTRIP:th.6">Th6</a></i>;<br/><a NAME="E4:30_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">k</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">c</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <b>by </b><i><a class="txt" href="#E3:30_1_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="nat_d.html#T8" title="NAT_D:th.8">NAT_D:8</a></i>;<br/><a NAME="E5:30_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">k1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="ref" href="pythtrip.html#T6" target="_self" title="PYTHTRIP:th.6">Th6</a></i>;<br/><a NAME="E6:30_1_1_1_1"/><b>then </b>
for <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 <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c8">k1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">n</font>
 <b>by </b><i><a class="txt" href="#E3:30_1_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="#E1:30_1_1_1_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><b>hence </b><a NAME="E7:30_1_1_1_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="#E1:30_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="pythtrip.html#D7" target="_self" title="PYTHTRIP:def.7">Def7</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><b>hence </b><a NAME="E7:30_1_1_1"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="pythtrip.html#D1" target="_self" title="PYTHTRIP:def.1">Def1</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>

<b>assume </b><a NAME="E2:30_1_1"/>
 ex <font color="Olive" title="b1">m</font>, <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="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="pythtrip.html#V3" title="PYTHTRIP:attr.3">simplified</a></span><br/>

<b>then consider </b><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">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> <b> such that </b><br/><a NAME="E4:30_1_1"/><i><font color="Green" title="E18">A4</font></i>: 
( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )
 ;<br/>
<a NAME="E5:30_1_1"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="#E4:30_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">k</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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pythtrip.html#D7" target="_self" title="PYTHTRIP:def.7">PYTHTRIP:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <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 <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c4">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">n</font> ) implies <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<b>assume </b><a NAME="E6:30_1_1"/>
for <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 <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c4">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E7:30_1_1"/><b>then </b>
( <font color="Maroon" title="c4">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c4">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n</font> )
 
<b>by </b><i><a class="txt" href="#E4:30_1_1"><i><font color="Green" title="E18">A4</font></i></a></i>;<br/>
<a NAME="E8:30_1_1"/><b>then </b>
<font color="Maroon" title="c4">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 1
 
<b>by </b><i><a class="txt" href="#E5:30_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:30_1_1"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="wsierp_1.html#T20" title="WSIERP_1:th.20">WSIERP_1:20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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