<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><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/>
<span class="kw">set </span><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="E1:86_1_1"/>
<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#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E2:86_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A4</font></span>: 
<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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E3:86_1_1"/>
<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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E3:86"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="ref" href="graph_2.html#D1" target="_self" title="GRAPH_2:def.1">Def1</a></span>;<br/>
<a NAME="E4:86_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A5</font></span>: 
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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a></span>;<br/>
<a NAME="E5:86_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_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> 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/>
<span class="kw">then </span><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:86_1_1"><span class="lab"><font color="Green" title="E62">A5</font></span></a></span>;<br/>
<span class="kw">set </span><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="xxreal_2.html#K2" title="XXREAL_2:func.2">min</a> <font color="Maroon" title="c6">X</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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#K2" title="NAT_1:func.2">+</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="E7:86_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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E8:86_1_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="xxreal_2.html#K2" title="XXREAL_2:func.2">min</a> <font color="Maroon" title="c6">X</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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="xxreal_2.html#K2" title="XXREAL_2:func.2">min</a> <font color="Maroon" title="c6">X</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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#K2" title="NAT_1:func.2">+</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">q</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">X</font> being 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>  st <br/>( <font color="Olive" title="b1">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> &amp; <font color="Maroon" title="c8">q</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="xxreal_2.html#K2" title="XXREAL_2:func.2">min</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">m</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">X</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <font color="Maroon" title="c8">q</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="xxreal_2.html#K2" title="XXREAL_2:func.2">min</a> <font color="Maroon" title="c6">X</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">m</font> )</span><br/>

<span class="kw">thus </span><a NAME="E10:86_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">q</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="xxreal_2.html#K2" title="XXREAL_2:func.2">min</a> <font color="Maroon" title="c6">X</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">m</font></span><br/>

<span class="kw">thus </span><a NAME="E11:86_1_1"/>
<font color="Maroon" title="c8">q</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="xxreal_2.html#K2" title="XXREAL_2:func.2">min</a> <font color="Maroon" title="c6">X</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
