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<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>





<b>set </b><font color="Maroon">lmnf</font> =  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K1">-cut</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">lf</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1"/>
( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) or  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> or  not <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or  not <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><b>suppose </b></a><a NAME="E1:17_1_1"/><i><font color="Green">E34</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/><div class="add"><a NAME="E2:17_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K1">-cut</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><a NAME="E3:17_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K1">-cut</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:17_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K1">-cut</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/><a NAME="E5:17_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K1">-cut</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:17_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K1">-cut</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E7:17_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K1">-cut</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E8:17_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K1">-cut</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2"><b>suppose </b></a><a NAME="E1:17_1_2"/>
(  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> or  not <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or  not <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/><div class="add"><a NAME="E2:17_1_2"/><b>then </b>
<font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K1">-cut</a> <font color="Maroon">f</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><a NAME="E3:17_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K1">-cut</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>hence </b><a NAME="E4:17_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K1">-cut</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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