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<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:2_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="int_2.html#R2">are_relative_prime</a> 
 ;<br/><b>let </b><font color="Maroon">p</font> be  <a href="int_2.html#V1">prime</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E2:2_1_1_1"/>
( <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> &amp; <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E3:2_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:2_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">p</font> being <a href="int_2.html#V1">prime</a> <a href="nat_1.html#NM1">Nat</a>  holds <br/>(  not <font color="Olive">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> or  not <font color="Olive">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> )
 ;<br/>

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="pythtrip.html#D1" target="_self">PYTHTRIP:def 1</a><br/>

<b>assume </b><a NAME="E3:2_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">m</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_2"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1"><b>suppose </b></a><a NAME="E1:2_1_1_2_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:2_1_1_2_1"/><b>then </b>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="int_2.html#T3">INT_2:3</a></i>;<br/><a NAME="E3:2_1_1_2_1"/><b>then </b>
( 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> &amp; 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T53">NAT_1:53</a></i>;<br/><b>hence </b><a NAME="E4:2_1_1_2_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="int_2.html#T44">INT_2:44</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_2"><b>suppose </b></a><a NAME="E1:2_1_1_2_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_2_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_3"><b>suppose </b></a><a NAME="E1:2_1_1_2_3"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:2_1_1_2_3"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <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/><b>then consider </b><font color="Maroon">p</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="E4:2_1_1_2_3"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">p</font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="int_2.html#T48">INT_2:48</a></i>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as  <a href="int_2.html#V1">prime</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E6:2_1_1_2_3"/>
( <font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> &amp; <font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/><b>hence </b><a NAME="E7:2_1_1_2_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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