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<b>set </b><font color="Maroon">Cut</font> = <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">F</font>;<br/>
<b>set </b><font color="Maroon">X</font> =  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">F</font></span>)</span>;<br/>
<a NAME="E1:4_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">F</font></span>)</span> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K4">INT</a> 
 
;<br/>
<a NAME="E2:4_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:4_1_1"/>
<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#K2">-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="E4:4_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
 <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#K2">-cut</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font>
 
;<br/>
<a NAME="E5:4_1_1"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&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="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E6:4_1_1"/><b>then </b>
not <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">F</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">X</font> =  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">F</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K4">INT</a>  <b>by </b><i>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>
<b>set </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p3">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-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="xcmplx_0.html#K6">-</a> 1;<br/>
<a NAME="E8:4_1_1"/>
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E9:4_1_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">F</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1 )
 
;<br/>
<a NAME="E10:4_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p3">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-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="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<b>take </b>
<font color="Maroon">q</font>
;<br/>

<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>thus </b><a NAME="E12:4_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E13:4_1_1"/>
<font color="Maroon">q</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 ;<br/>


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