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<b>let </b><font color="Maroon" title="c6">m</font>, <font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D1</font>
 <b>and </b><br/><a NAME="E2:62_1_1_1"/><i><font color="Green" title="E43">A2</font></i>: 
( <font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D2</font> &amp; <font color="Maroon" title="c3">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c6">m</font> )
 <b>and </b><br/><a NAME="E3:62_1_1_1"/><i><font color="Green" title="E44">A3</font></i>: 
( <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D2</font> &amp; <font color="Maroon" title="c3">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">n</font> )
 ;<br/><b>assume </b><a NAME="E4:62_1_1_1"/><i><font color="Green" title="E45">A4</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">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" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n</font> or <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font> )
 </span><b>by </b><i><a class="txt" href="integra1.html#E4:62_1_1_1"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">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="txt" href="integra1.html#E2:62_1_1_1"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="integra1.html#E3:62_1_1_1"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">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" title="c7">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">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="txt" href="integra1.html#E2:62_1_1_1"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="integra1.html#E3:62_1_1_1"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">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" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D1</font>
 <b>and </b><br/><a NAME="E3:62_1_1"/><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>and </b><br/><a NAME="E4:62_1_1"/><i><font color="Green" title="E44">A6</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<b>thus </b><a NAME="E5:62_1_1"/>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="integra1.html#E3:62_1_1"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="integra1.html#E4:62_1_1"><i><font color="Green" title="E44">A6</font></i></a></i>;<br/>


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