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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Olive" title="b1">p</font> is   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">p</font> is   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c3">m</font>,<font color="Maroon" title="c4">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">p</font> is   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font></span><br/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:12_1"/>
(  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> or  not <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> or <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">n</font> or ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1"><span class="kw">suppose </span></a><a NAME="E1:12_1_1"/>
(  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> or  not <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> or <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">n</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="c3">m</font>,<font color="Maroon" title="c4">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">p</font> is   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font></span><br/><div class="add"><a NAME="E2:12_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:12_1_1"/>
<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">p</font> is   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_1.html#T14" title="GRAPH_1:th.14">GRAPH_1:14</a></span>; <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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2"><span class="kw">suppose </span></a><a NAME="E1:12_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</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="c3">m</font>,<font color="Maroon" title="c4">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">p</font> is   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">p</font> as    <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#T44" title="GRAPH_2:th.44">GRAPH_2:44</a></span>;<br/><span class="kw">assume </span><a NAME="E3:12_1_2"/>
<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">p</font> is  not   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:12_1_2"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">and </span><br/><a NAME="E6:12_1_2"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">and </span><br/><a NAME="E7:12_1_2"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">and </span><br/><a NAME="E8:12_1_2"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_1.html#D14" title="GRAPH_1:def.14">GRAPH_1:def 14</a></span>;<br/><a NAME="E9:12_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12_1_2"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E6:12_1_2"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:12_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_1_2"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">kb</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E12:12_1_2"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
( <font color="Maroon" title="c8">kb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">kb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font> )
 <span class="kw">and </span><br/><a NAME="E13:12_1_2"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
<font color="Maroon" title="c8">kb</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">and </span><br/><a NAME="E14:12_1_2"/>
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">kb</font>
 <span class="kw">and </span><br/><a NAME="E15:12_1_2"/>
<font color="Maroon" title="c8">kb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#T2" target="_self" title="GRAPH_3:th.2">Th2</a></span>;<br/><a NAME="E16:12_1_2"/>
<font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_1_2"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E7:12_1_2"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E17:12_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12_1_2"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">ka</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E19:12_1_2"/><span class="lab"><font color="Green" title="E13">A7</font></span>: 
( <font color="Maroon" title="c9">ka</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">ka</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> )
 <span class="kw">and </span><br/><a NAME="E20:12_1_2"/><span class="lab"><font color="Green" title="E14">A8</font></span>: 
<font color="Maroon" title="c9">ka</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">and </span><br/><a NAME="E21:12_1_2"/>
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">ka</font>
 <span class="kw">and </span><br/><a NAME="E22:12_1_2"/>
<font color="Maroon" title="c9">ka</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#T2" target="_self" title="GRAPH_3:th.2">Th2</a></span>;<br/><a NAME="E23:12_1_2"/>
<font color="Maroon" title="c9">ka</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">kb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_1_2"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E19:12_1_2"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="txt" href="#E12:12_1_2"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E24:12_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_1_2"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E20:12_1_2"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="txt" href="#E13:12_1_2"><span class="lab"><font color="Green" title="E12">A6</font></span></a></span>; <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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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