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<div class="add">

<b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">m</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:53"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">m</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>thus </b><a NAME="E2:53"/>
( <span class="p1">(<span class="default"><a href="finseq_2.html#K1">idseq</a> <font color="Maroon">n</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_2.html#K1">idseq</a> <font color="Maroon">m</font> implies <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1"/>
<span class="p1">(<span class="default"><a href="finseq_2.html#K1">idseq</a> <font color="Maroon">n</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_2.html#K1">idseq</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:53_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_2.html#K1">idseq</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_2.html#K1">idseq</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T3" target="_self">Th3</a>, <a class="ref" href="trees_1.html#T2">TREES_1:2</a></i>;<br/>
<a NAME="E3:53_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_2.html#K1">idseq</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T55">FINSEQ_2:55</a></i>;<br/>
<b>hence </b><a NAME="E4:53_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T55">FINSEQ_2:55</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:53"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E4:53"/><b>then </b>
 ex <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">j</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>hence </b><a NAME="E5:53"/>
<span class="p1">(<span class="default"><a href="finseq_2.html#K1">idseq</a> <font color="Maroon">n</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_2.html#K1">idseq</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T2" target="_self">Th2</a></i>;<br/>


</div>
