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<b>set </b><font color="Maroon" title="c4">Cut</font> = <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c1">F</font>;<br/>
<b>set </b><font color="Maroon" title="c5">X</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c1">F</font></span>)</span>;<br/>
<a NAME="E1:84_1_1"/><i><font color="Green" title="E59">A2</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c1">F</font></span>)</span> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
;<br/>
<a NAME="E2:84_1_1"/><i><font color="Green" title="E60">A3</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="graph_2.html#E1:84"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:84_1_1"/><i><font color="Green" title="E61">A4</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="graph_2.html#E1:84"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="graph_2.html#D1" target="_self" title="GRAPH_2:def.1">Def1</a></i>;<br/>
<a NAME="E4:84_1_1"/>
<font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="txt" href="graph_2.html#E2:84_1_1"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E5:84_1_1"/><b>then </b>
not <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c1">F</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="graph_2.html#E3:84_1_1"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">X</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c1">F</font></span>)</span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <b>by </b><i><a class="txt" href="graph_2.html#E1:84_1_1"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">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" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c6">X</font></span>)</span> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/>
<a NAME="E7:84_1_1"/>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="graph_2.html#E1:84"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E8:84_1_1"/><b>then </b>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c6">X</font></span>)</span> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">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" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c6">X</font></span>)</span> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">q</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c6">X</font>
;<br/>

<b>thus </b><a NAME="E10:84_1_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c1">F</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E11:84_1_1"/>
<font color="Maroon" title="c8">q</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c6">X</font></span>)</span> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">m</font>
 ;<br/>


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