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<b>let </b><font color="Maroon">v</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:21"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">v</font> is <a href="goboard1.html#V1">increasing</a>
 ;<br/>

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



<b>assume </b><a NAME="E2:21"/><i><font color="Green">E51</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">v</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</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="21_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_1"/>
<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="E2:21_1_1_1"/>
<font color="Maroon">v</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>
 <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="21_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_2"/>
<font color="Maroon">v</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>
 <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="E4:21"/>
<font color="Maroon">v</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>
 ;<br/>


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