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<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/><b>let </b><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n</font> be    <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> ;<br/>





<b>set </b><font color="Maroon" title="c4">c</font> = <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font> <a href="graph_2.html#K1" title="GRAPH_2:func.1">-cut</a> <font color="Maroon" title="c1">p</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green" title="E14">A1</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <b>and </b><br/><a NAME="E3:16"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c3">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="c1">p</font>
 ;<br/>

<a NAME="E4:16"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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#E2:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E5:16"/><b>then </b><i><font color="Green" title="E18">A5</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#K1" title="GRAPH_2:func.1">-cut</a> <font color="Maroon" title="c1">p</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#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="graph_2.html#E3:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green" title="E8">Lm2</font></i></a></i>;<br/>
<div><i><font color="Green" title="E19">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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#K1" title="GRAPH_2:func.1">-cut</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:16_1"/><b>then </b>
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
 <b>by </b><i><a class="txt" href="graph_2.html#E2:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="graph_2.html#E5:16"><i><font color="Green" title="E18">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E3:16_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:16"/><i><font color="Green" title="E20">A7</font></i>: 
0 <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> <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#K1" title="GRAPH_2:func.1">-cut</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
;<br/>
<a NAME="E8:16"/>
( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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#K1" title="GRAPH_2:func.1">-cut</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="graph_2.html#E6:16"><i><font color="Green" title="E19">A6</font></i></a></i>;<br/>
<b>hence </b><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#K1" title="GRAPH_2:func.1">-cut</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="graph_2.html#E3:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="graph_2.html#E4:16"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green" title="E8">Lm2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">m</font>

;<br/><br/>

<a NAME="E10:16"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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> <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#K1" title="GRAPH_2:func.1">-cut</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_2.html#E6:16"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="graph_2.html#E7:16"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">i</font> being    <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> such that </b><br/><a NAME="E12:16"/><i><font color="Green" title="E21">A8</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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#K1" title="GRAPH_2:func.1">-cut</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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#K1" title="GRAPH_2:func.1">-cut</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="ref" href="graph_2.html#T1" target="_self" title="GRAPH_2:th.1">Th1</a></i>;<br/>
<a NAME="E13:16"/>
<font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="graph_2.html#E5:16"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="graph_2.html#E12:16"><i><font color="Green" title="E21">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:16"/>
<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#K1" title="GRAPH_2:func.1">-cut</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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#K1" title="GRAPH_2:func.1">-cut</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="graph_2.html#E3:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="graph_2.html#E4:16"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="graph_2.html#E12:16"><i><font color="Green" title="E21">A8</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green" title="E8">Lm2</font></i></a></i>;<br/>


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