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

<div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:62_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font>
 <b>and </b><br/><a NAME="E2:62_1_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> )
 <b>and </b><br/><a NAME="E3:62_1_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/><b>assume </b><a NAME="E4:62_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_1_1_1_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> or <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:62_1_1_1_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:62_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:62_1_1_1_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:62_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:62_1_1_1"/>
contradiction
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E2:62_1_1"/>
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font>
 <b>and </b><br/><a NAME="E3:62_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E4:62_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>thus </b><a NAME="E5:62_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/>


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