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<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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">m</font>, <font color="Olive" title="b2">n</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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">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> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">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 <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="Olive" title="b1">m</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">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="Olive" title="b1">m</font>,<font color="Olive" title="b2">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="Olive" title="b2">n</font> )</span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> &amp; <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> &amp; <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> implies ( <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 <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="c2">m</font> &amp; <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> ) )</span><br/>





<span class="kw">set </span><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/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">and </span><br/><a NAME="E2:16"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:16"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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 <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="c2">m</font> &amp; <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> )</span><br/>

<a NAME="E4:16"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<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#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E5:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<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#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="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="graph_2.html#E8"><span class="lab"><font color="Green" title="E8">Lm2</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E19">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:16_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</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>
<a NAME="E7:16"/><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> 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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E9:16"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">and </span><br/><a NAME="E10:16"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E11:16"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
 <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#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#T1" target="_self" title="GRAPH_2:th.1">Th1</a></span>;<br/>
<a NAME="E12:16"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
;<br/>
<span class="kw">hence </span><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#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="graph_2.html#E8"><span class="lab"><font color="Green" title="E8">Lm2</font></span></a></span>
<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/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E14:16"/>
<font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E21">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:16"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="graph_2.html#E8"><span class="lab"><font color="Green" title="E8">Lm2</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>
