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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">vs</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">c</font> be  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font>;<br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E19">A1</font></i>: 
( not <font color="Maroon" title="c3">c</font> is <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a> &amp; <font color="Maroon" title="c2">vs</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Maroon" title="c3">c</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">n</font>, <font color="Maroon" title="c5">m</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> <b> such that </b><br/><a NAME="E3:31"/><i><font color="Green" title="E20">A2</font></i>: 
( 1 <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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c5">m</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">vs</font> &amp; <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> &amp; ( <font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or <font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font> ) )
 <b>by </b><i><a class="ref" href="graph_4.html#D7" target="_self" title="GRAPH_4:def.7">Def7</a></i>;<br/>
<a NAME="E4:31"/><i><font color="Green" title="E21">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="graph_4.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="graph_4.html#D5" target="_self" title="GRAPH_4:def.5">Def5</a></i>;<br/>
<a NAME="E5:31"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">n1</font> = <font color="Maroon" title="c4">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 as    <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>  ;<br/>
<a NAME="E7:31"/><i><font color="Green" title="E23">A5</font></i>: 
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="c4">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E8:31"/><b>then </b><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<i><font color="Green" title="E25">A7</font></i>: <font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<b>by </b><i><a class="txt" href="graph_4.html#E7:31"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c4">n</font>

;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2"/>
( ( <font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font> ) or ( <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font> ) or ( <font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font> ) )
 </span><b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1"><b>suppose </b></a><a NAME="E1:31_2_1"/><i><font color="Green" title="E26">A8</font></i>: 
( <font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font> )
 ;<br/><div class="add"><a NAME="E2:31_2_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:31_2_1"/><b>then </b>
1 <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="c4">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:31_2_1"/><b>then </b><i><font color="Green" title="E27">A9</font></i>: 
<span class="p1">(<span class="default">1 <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> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E5:31_2_1"/>
<font color="Maroon" title="c4">n</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> <font color="Maroon" title="c2">vs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:31_2_1"/><b>then </b><i><font color="Green" title="E28">A10</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <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> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">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="E7:31_2_1"/><i><font color="Green" title="E29">A11</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E8:31_2_1"/><i><font color="Green" title="E30">A12</font></i>: 
<font color="Maroon" title="c5">m</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> <font color="Maroon" title="c2">vs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graph_4.html#E1:31_2_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E9:31_2_1"/><b>then </b><i><font color="Green" title="E31">A13</font></i>: 
<font color="Maroon" title="c5">m</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="c3">c</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:31_2_1"/><i><font color="Green" title="E32">A14</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c5">m</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="c3">c</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</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="c3">c</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_1"><i><font color="Green" title="E30">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">c1</font> = 1,<font color="Maroon" title="c6">n1</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">c</font> as  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="graph_4.html#T13" target="_self" title="GRAPH_4:th.13">Th13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">c2</font> = <font color="Maroon" title="c5">m</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">c</font> as  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="graph_4.html#T13" target="_self" title="GRAPH_4:th.13">Th13</a></i>;<br/><b>set </b><font color="Maroon" title="c9">pp</font> = 1,<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">vs</font>;<br/><b>set </b><font color="Maroon" title="c10">p2</font> = <font color="Maroon" title="c5">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font>;<br/><b>set </b><font color="Maroon" title="c11">p2'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font>;<br/><b>reconsider </b><font color="Maroon" title="c12">pp</font> = 1,<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">vs</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font> ;<br/><b>reconsider </b><font color="Maroon" title="c13">p2</font> = <font color="Maroon" title="c5">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font> ;<br/><a NAME="E15:31_2_1"/><i><font color="Green" title="E33">A15</font></i>: 
( 1 <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">vs</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E16:31_2_1"/><i><font color="Green" title="E34">A16</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <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> <font color="Maroon" title="c2">vs</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="graph_4.html#D5" target="_self" title="GRAPH_4:def.5">Def5</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E17:31_2_1"/><i><font color="Green" title="E35">A17</font></i>: 
<font color="Maroon" title="c12">pp</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Maroon" title="c7">c1</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="graph_4.html#E9:31"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="graph_4.html#T14" target="_self" title="GRAPH_4:th.14">Th14</a></i>;<br/><a NAME="E18:31_2_1"/><i><font color="Green" title="E36">A18</font></i>: 
<font color="Maroon" title="c13">p2</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Maroon" title="c8">c2</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="graph_4.html#T14" target="_self" title="GRAPH_4:th.14">Th14</a></i>;<br/><a NAME="E19:31_2_1"/><i><font color="Green" title="E37">A19</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">pp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E17:31_2_1"><i><font color="Green" title="E35">A17</font></i></a>, <a class="ref" href="graph_4.html#D5" target="_self" title="GRAPH_4:def.5">Def5</a></i>;<br/><a NAME="E20:31_2_1"/><i><font color="Green" title="E38">A20</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">c2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E18:31_2_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="ref" href="graph_4.html#D5" target="_self" title="GRAPH_4:def.5">Def5</a></i>;<br/><a NAME="E21:31_2_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="c5">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E22:31_2_1"/><b>then </b>
<font color="Maroon" title="c5">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">m1</font> = <font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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>  ;<br/><a NAME="E24:31_2_1"/><i><font color="Green" title="E39">A21</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E25:31_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">c2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></i>;<br/><a NAME="E26:31_2_1"/><b>then </b><i><font color="Green" title="E40">A22</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ;<br/><a NAME="E27:31_2_1"/><i><font color="Green" title="E41">A23</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E28:31_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">c2</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">c2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E26:31_2_1"><i><font color="Green" title="E40">A22</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c15">lc21</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">c2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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>  ;<br/><a NAME="E30:31_2_1"/><i><font color="Green" title="E42">A24</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">lc21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">c2</font></span>)</span>
 ;<br/><a NAME="E31:31_2_1"/><i><font color="Green" title="E43">A25</font></i>: 
( 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 &amp; 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> <font color="Maroon" title="c13">p2</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_1"><i><font color="Green" title="E38">A20</font></i></a></i>;<br/><a NAME="E32:31_2_1"/><b>then </b><i><font color="Green" title="E44">A26</font></i>: 
( 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 &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 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> <font color="Maroon" title="c13">p2</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><i><font color="Green" title="E45">A27</font></i>: <font color="Maroon" title="c13">p2</font> = 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">p2</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c13">p2</font>
<b>by </b><i><a class="ref" href="graph_2.html#T7" title="GRAPH_2:th.7">GRAPH_2:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1 <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c13">p2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">p2</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c13">p2</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E32:31_2_1"><i><font color="Green" title="E44">A26</font></i></a>, <a class="ref" href="graph_2.html#T8" title="GRAPH_2:th.8">GRAPH_2:8</a></i>
;<br/><a NAME="E34:31_2_1"/>
<font color="Maroon" title="c14">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E24:31_2_1"><i><font color="Green" title="E39">A21</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E35:31_2_1"/><b>then </b><i><font color="Green" title="E46">A28</font></i>: 
<font color="Maroon" title="c13">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">m</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="graph_2.html#T8" title="GRAPH_2:th.8">GRAPH_2:8</a></i>;<br/>1,1 <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c13">p2</font> = 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="graph_4.html#E32:31_2_1"><i><font color="Green" title="E44">A26</font></i></a>, <a class="ref" href="graph_2.html#T6" title="GRAPH_2:th.6">GRAPH_2:6</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="txt" href="graph_4.html#E31:31_2_1"><i><font color="Green" title="E43">A25</font></i></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c5">m</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font>
<b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="graph_2.html#T6" title="GRAPH_2:th.6">GRAPH_2:6</a></i>
;<br/><a NAME="E37:31_2_1"/><b>then </b><i><font color="Green" title="E47">A29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">p2</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c13">p2</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E33:31_2_1"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="graph_4.html#E35:31_2_1"><i><font color="Green" title="E46">A28</font></i></a>, <a class="ref" href="finseq_1.html#T46" title="FINSEQ_1:th.46">FINSEQ_1:46</a></i>;<br/><b>set </b><font color="Maroon" title="c16">domfc</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) or ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c3">c</font> ) ) </span>}</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/><b>consider </b><font color="Maroon" title="c17">fc</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E39:31_2_1"/><i><font color="Green" title="E48">A30</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c17">fc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) or ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c3">c</font> ) ) </span>}</span> 
 <b>and </b><br/><a NAME="E40:31_2_1"/><i><font color="Green" title="E49">A31</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) or ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</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="c3">c</font> ) ) </span>}</span>  holds <br/><font color="Maroon" title="c17">fc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E41:31_2_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) or ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c3">c</font> ) ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_1_3"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) or ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c3">c</font> ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c19">kk</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> <b> such that </b><br/><a NAME="E3:31_2_1_3"/><i><font color="Green" title="E50">A32</font></i>: 
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">kk</font> &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">kk</font> &amp; <font color="Maroon" title="c19">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) or ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">kk</font> &amp; <font color="Maroon" title="c19">kk</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="c3">c</font> ) ) )
 ;<br/>
<a NAME="E4:31_2_1_3"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">kk</font> &amp; <font color="Maroon" title="c19">kk</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="c3">c</font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_3"><i><font color="Green" title="E50">A32</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2_1_3"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_3"><i><font color="Green" title="E50">A32</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c18">fc</font> = <font color="Maroon" title="c17">fc</font> as   <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a> <b>by </b><i><a class="txt" href="graph_4.html#E39:31_2_1"><i><font color="Green" title="E48">A30</font></i></a>, <a class="ref" href="finseq_1.html#D12" title="FINSEQ_1:def.12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E43:31_2_1"/><i><font color="Green" title="E50">A33</font></i>: 
<font color="Maroon" title="c18">fc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_1_4"/><i><font color="Green" title="E51">A34</font></i>: 
<font color="Maroon" title="c19">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">fc</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c20">x</font>, <font color="Maroon" title="c21">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_2_1_4"/><i><font color="Green" title="E52">A35</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c21">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">p</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:31_2_1_4"/><i><font color="Green" title="E53">A36</font></i>: 
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c18">fc</font> &amp; <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">fc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_1_4"><i><font color="Green" title="E51">A34</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_4"><i><font color="Green" title="E52">A35</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">kk</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> <b> such that </b><br/><a NAME="E6:31_2_1_4"/><i><font color="Green" title="E54">A37</font></i>: 
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">kk</font> &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">kk</font> &amp; <font color="Maroon" title="c22">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) or ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">kk</font> &amp; <font color="Maroon" title="c22">kk</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="c3">c</font> ) ) )
 <b>by </b><i><a class="txt" href="graph_4.html#E39:31_2_1"><i><font color="Green" title="E48">A30</font></i></a></i>;<br/>
<a NAME="E7:31_2_1_4"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">kk</font> &amp; <font color="Maroon" title="c22">kk</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="c3">c</font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_4"><i><font color="Green" title="E54">A37</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:31_2_1_4"/><b>then </b><i><font color="Green" title="E55">A38</font></i>: 
<font color="Maroon" title="c20">x</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="c3">c</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_1_4"><i><font color="Green" title="E54">A37</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:31_2_1_4"/>
<font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">kk</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E39:31_2_1"><i><font color="Green" title="E48">A30</font></i></a>, <a class="txt" href="graph_4.html#E40:31_2_1"><i><font color="Green" title="E49">A31</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_4"><i><font color="Green" title="E53">A36</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_4"><i><font color="Green" title="E54">A37</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:31_2_1_4"/>
<font color="Maroon" title="c19">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_4"><i><font color="Green" title="E52">A35</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_4"><i><font color="Green" title="E54">A37</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_1_4"><i><font color="Green" title="E55">A38</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c19">fc</font> = <font color="Maroon" title="c18">fc</font> as    <a href="graph_2.html#M1" title="GRAPH_2:mode.1">FinSubsequence</a> of <font color="Maroon" title="c3">c</font> <b>by </b><i><a class="ref" href="graph_2.html#D5" title="GRAPH_2:def.5">GRAPH_2:def 5</a></i>;<br/><b>take </b>
<font color="Maroon" title="c19">fc</font>
;<br/><b>set </b><font color="Maroon" title="c20">domfvs</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) or ( <font color="Maroon" title="c5">m</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="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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">vs</font> ) ) </span>}</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/><b>consider </b><font color="Maroon" title="c21">fvs</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E46:31_2_1"/><i><font color="Green" title="E51">A39</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c21">fvs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) or ( <font color="Maroon" title="c5">m</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="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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">vs</font> ) ) </span>}</span> 
 <b>and </b><br/><a NAME="E47:31_2_1"/><i><font color="Green" title="E52">A40</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) or ( <font color="Maroon" title="c5">m</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="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</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">vs</font> ) ) </span>}</span>  holds <br/><font color="Maroon" title="c21">fvs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E48:31_2_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) or ( <font color="Maroon" title="c5">m</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="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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">vs</font> ) ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_1_5"/>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) or ( <font color="Maroon" title="c5">m</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="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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">vs</font> ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c23">kk</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> <b> such that </b><br/><a NAME="E3:31_2_1_5"/><i><font color="Green" title="E53">A41</font></i>: 
( <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">kk</font> &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">kk</font> &amp; <font color="Maroon" title="c23">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) or ( <font color="Maroon" title="c5">m</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="c23">kk</font> &amp; <font color="Maroon" title="c23">kk</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">vs</font> ) ) )
 ;<br/>
<a NAME="E4:31_2_1_5"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E5:31_2_1_5"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">kk</font> &amp; <font color="Maroon" title="c23">kk</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">vs</font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E15:31_2_1"><i><font color="Green" title="E33">A15</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_5"><i><font color="Green" title="E53">A41</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:31_2_1_5"/>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_5"><i><font color="Green" title="E53">A41</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c22">fvs</font> = <font color="Maroon" title="c21">fvs</font> as   <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a> <b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_1"><i><font color="Green" title="E51">A39</font></i></a>, <a class="ref" href="finseq_1.html#D12" title="FINSEQ_1:def.12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E50:31_2_1"/><i><font color="Green" title="E53">A42</font></i>: 
<font color="Maroon" title="c22">fvs</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">vs</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_1_6"/><i><font color="Green" title="E54">A43</font></i>: 
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">fvs</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c24">x</font>, <font color="Maroon" title="c25">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_2_1_6"/><i><font color="Green" title="E55">A44</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c25">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:31_2_1_6"/><i><font color="Green" title="E56">A45</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c22">fvs</font> &amp; <font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">fvs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_1_6"><i><font color="Green" title="E54">A43</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_6"><i><font color="Green" title="E55">A44</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c26">kk</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> <b> such that </b><br/><a NAME="E6:31_2_1_6"/><i><font color="Green" title="E57">A46</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">kk</font> &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">kk</font> &amp; <font color="Maroon" title="c26">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) or ( <font color="Maroon" title="c5">m</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="c26">kk</font> &amp; <font color="Maroon" title="c26">kk</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">vs</font> ) ) )
 <b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_1"><i><font color="Green" title="E51">A39</font></i></a></i>;<br/>
<a NAME="E7:31_2_1_6"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E8:31_2_1_6"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">kk</font> &amp; <font color="Maroon" title="c26">kk</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">vs</font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E15:31_2_1"><i><font color="Green" title="E33">A15</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_6"><i><font color="Green" title="E57">A46</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:31_2_1_6"/><b>then </b><i><font color="Green" title="E58">A47</font></i>: 
<font color="Maroon" title="c24">x</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">vs</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_1_6"><i><font color="Green" title="E57">A46</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:31_2_1_6"/>
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">kk</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_1"><i><font color="Green" title="E51">A39</font></i></a>, <a class="txt" href="graph_4.html#E47:31_2_1"><i><font color="Green" title="E52">A40</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_6"><i><font color="Green" title="E56">A45</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_6"><i><font color="Green" title="E57">A46</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:31_2_1_6"/>
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">vs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_6"><i><font color="Green" title="E55">A44</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_6"><i><font color="Green" title="E57">A46</font></i></a>, <a class="txt" href="graph_4.html#E9:31_2_1_6"><i><font color="Green" title="E58">A47</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c23">fvs</font> = <font color="Maroon" title="c22">fvs</font> as    <a href="graph_2.html#M1" title="GRAPH_2:mode.1">FinSubsequence</a> of <font color="Maroon" title="c2">vs</font> <b>by </b><i><a class="ref" href="graph_2.html#D5" title="GRAPH_2:def.5">GRAPH_2:def 5</a></i>;<br/><b>take </b>
<font color="Maroon" title="c23">fvs</font>
;<br/><a NAME="E52:31_2_1"/><i><font color="Green" title="E54">A48</font></i>: 
( <font color="Maroon" title="c13">p2</font> <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="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c12">pp</font> <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> <font color="Maroon" title="c12">pp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E15:31_2_1"><i><font color="Green" title="E33">A15</font></i></a>, <a class="txt" href="graph_4.html#E16:31_2_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="ref" href="graph_2.html#T12" title="GRAPH_2:th.12">GRAPH_2:12</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c24">c'</font> = <font color="Maroon" title="c7">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c8">c2</font> as  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graph_4.html#E17:31_2_1"><i><font color="Green" title="E35">A17</font></i></a>, <a class="txt" href="graph_4.html#E18:31_2_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="ref" href="graph_4.html#T15" target="_self" title="GRAPH_4:th.15">Th15</a></i>;<br/><b>take </b>
<font color="Maroon" title="c24">c'</font>
;<br/><b>take </b><font color="Maroon" title="c25">p1</font> = <font color="Maroon" title="c12">pp</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c13">p2</font>;<br/><a NAME="E54:31_2_1"/><i><font color="Green" title="E55">A49</font></i>: 
<font color="Maroon" title="c25">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">pp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E37:31_2_1"><i><font color="Green" title="E47">A29</font></i></a>, <a class="ref" href="graph_2.html#D2" title="GRAPH_2:def.2">GRAPH_2:def 2</a></i>;<br/><a NAME="E55:31_2_1"/><i><font color="Green" title="E56">A50</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_1"><i><font color="Green" title="E29">A11</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green" title="E17">Lm2</font></i></a></i>;<br/><a NAME="E56:31_2_1"/><b>then </b><i><font color="Green" title="E57">A51</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/><a NAME="E57:31_2_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">n</font>
 ;<br/><a NAME="E58:31_2_1"/><b>then </b><i><font color="Green" title="E58">A52</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 <b>by </b><i><a class="txt" href="graph_4.html#E5:31"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="xreal_1.html#T60" title="XREAL_1:th.60">XREAL_1:60</a></i>;<br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">c'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E26:31_2_1"><i><font color="Green" title="E40">A22</font></i></a>, <a class="txt" href="graph_4.html#E55:31_2_1"><i><font color="Green" title="E56">A50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span>

;<br/><b>hence </b><a NAME="E60:31_2_1"/><i><font color="Green" title="E59">A53</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">c'</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> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E58:31_2_1"><i><font color="Green" title="E58">A52</font></i></a>, <a class="ref" href="real_2.html#T173" title="REAL_2:th.173">REAL_2:173</a></i>;<br/><b>thus </b><a NAME="E61:31_2_1"/>
<font color="Maroon" title="c25">p1</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Maroon" title="c24">c'</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graph_4.html#E17:31_2_1"><i><font color="Green" title="E35">A17</font></i></a>, <a class="txt" href="graph_4.html#E18:31_2_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="graph_4.html#E52:31_2_1"><i><font color="Green" title="E54">A48</font></i></a>, <a class="ref" href="graph_4.html#T16" target="_self" title="GRAPH_4:th.16">Th16</a></i>;<br/><a NAME="E62:31_2_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">c'</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="graph_4.html#D5" target="_self" title="GRAPH_4:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E63:31_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">p1</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> <font color="Maroon" title="c2">vs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="graph_4.html#E60:31_2_1"><i><font color="Green" title="E59">A53</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E64:31_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c1</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E65:31_2_1"/><b>then </b>
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> <font color="Maroon" title="c12">pp</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E17:31_2_1"><i><font color="Green" title="E35">A17</font></i></a>, <a class="ref" href="graph_4.html#D5" target="_self" title="GRAPH_4:def.5">Def5</a></i>;<br/><a NAME="E66:31_2_1"/><b>then </b>
<font color="Maroon" title="c25">p1</font> <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="c12">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="ref" href="graph_2.html#T14" title="GRAPH_2:th.14">GRAPH_2:14</a></i>;<br/><b>hence </b><a NAME="E67:31_2_1"/>
<font color="Maroon" title="c2">vs</font> <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="c25">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E15:31_2_1"><i><font color="Green" title="E33">A15</font></i></a>, <a class="ref" href="graph_2.html#T12" title="GRAPH_2:th.12">GRAPH_2:12</a></i>;<br/><a NAME="E68:31_2_1"/><i><font color="Green" title="E60">A54</font></i>: 
<font color="Maroon" title="c13">p2</font> <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> <font color="Maroon" title="c13">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="ref" href="graph_2.html#T12" title="GRAPH_2:th.12">GRAPH_2:12</a></i>;<br/><a NAME="E69:31_2_1"/>
<font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E70:31_2_1"/><b>then </b>
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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E71:31_2_1"/><b>then </b>
1 <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> <font color="Maroon" title="c13">p2</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_1"><i><font color="Green" title="E38">A20</font></i></a>, <a class="txt" href="graph_4.html#E26:31_2_1"><i><font color="Green" title="E40">A22</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E72:31_2_1"/>
<font color="Maroon" title="c2">vs</font> <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> <font color="Maroon" title="c2">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">p1</font> <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> <font color="Maroon" title="c25">p1</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="graph_4.html#E68:31_2_1"><i><font color="Green" title="E60">A54</font></i></a>, <a class="ref" href="graph_2.html#T16" title="GRAPH_2:th.16">GRAPH_2:16</a></i>;<br/><b>set </b><font color="Maroon" title="c26">DL</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c27">DR</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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="c3">c</font> ) </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c28">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_1_8_1"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) or ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c3">c</font> ) ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c29">k</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> <b> such that </b><br/><a NAME="E3:31_2_1_8_1"/><i><font color="Green" title="E61">A55</font></i>: 
( <font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">k</font> &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">k</font> &amp; <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) or ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">k</font> &amp; <font color="Maroon" title="c29">k</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="c3">c</font> ) ) )
 ;<br/><a NAME="E4:31_2_1_8_1"/>
( <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span>  or <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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="c3">c</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_8_1"><i><font color="Green" title="E61">A55</font></i></a></i>;<br/><b>hence </b><a NAME="E5:31_2_1_8_1"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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="c3">c</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_1_8"/><i><font color="Green" title="E61">A56</font></i>: 
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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="c3">c</font> ) </span>}</span> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_8_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_8_2"/>
( <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span>  or <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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="c3">c</font> ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_1_8"><i><font color="Green" title="E61">A56</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_8_2_1"><b>suppose </b></a><a NAME="E1:31_2_1_8_2_1"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c29">k</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> <b> such that </b><br/><a NAME="E3:31_2_1_8_2_1"/><i><font color="Green" title="E62">A57</font></i>: 
( <font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">k</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">k</font> &amp; <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> )
 ;<br/><b>thus </b><a NAME="E4:31_2_1_8_2_1"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) or ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c3">c</font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_8_2_1"><i><font color="Green" title="E62">A57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_8_2_2"><b>suppose </b></a><a NAME="E1:31_2_1_8_2_2"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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="c3">c</font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c29">k</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> <b> such that </b><br/><a NAME="E3:31_2_1_8_2_2"/><i><font color="Green" title="E62">A58</font></i>: 
( <font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">k</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">k</font> &amp; <font color="Maroon" title="c29">k</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="c3">c</font> )
 ;<br/><b>thus </b><a NAME="E4:31_2_1_8_2_2"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) or ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c3">c</font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_8_2_2"><i><font color="Green" title="E62">A58</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E74:31_2_1"/><b>then </b><i><font color="Green" title="E61">A59</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) or ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c3">c</font> ) ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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="c3">c</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E75:31_2_1"/><i><font color="Green" title="E62">A60</font></i>: 
( <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> &amp; <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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="c3">c</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_9"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>
<div class="add"><b>let </b><font color="Maroon" title="c28">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:31_2_1_9_1"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c29">k</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> <b> such that </b><br/><a NAME="E3:31_2_1_9_1"/><i><font color="Green" title="E63">A61</font></i>: 
( <font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">k</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">k</font> &amp; <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> )
 ;<br/><a NAME="E4:31_2_1_9_1"/>
<font color="Maroon" title="c29">k</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="c3">c</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_9_1"><i><font color="Green" title="E63">A61</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:31_2_1_9_1"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_9_1"><i><font color="Green" title="E63">A61</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c28">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:31_2_1_9"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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="c3">c</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c29">k</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> <b> such that </b><br/><a NAME="E4:31_2_1_9"/><i><font color="Green" title="E63">A62</font></i>: 
( <font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">k</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">k</font> &amp; <font color="Maroon" title="c29">k</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="c3">c</font> )
 ;<br/>
<a NAME="E5:31_2_1_9"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">k</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_9"><i><font color="Green" title="E63">A62</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:31_2_1_9"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_9"><i><font color="Green" title="E63">A62</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c28">DL</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">DR</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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="c3">c</font> ) </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="graph_4.html#E75:31_2_1"><i><font color="Green" title="E62">A60</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c30">i</font>, <font color="Maroon" title="c31">j</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> ;<br/><b>assume </b><a NAME="E1:31_2_1_10"/>
<font color="Maroon" title="c30">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">DL</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c32">k</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> <b> such that </b><br/><a NAME="E3:31_2_1_10"/><i><font color="Green" title="E63">A63</font></i>: 
( <font color="Maroon" title="c32">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">k</font> &amp; <font color="Maroon" title="c32">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> )
 ;<br/><b>assume </b><a NAME="E4:31_2_1_10"/>
<font color="Maroon" title="c31">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">DR</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c33">l</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> <b> such that </b><br/><a NAME="E6:31_2_1_10"/><i><font color="Green" title="E64">A64</font></i>: 
( <font color="Maroon" title="c33">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">j</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c33">l</font> &amp; <font color="Maroon" title="c33">l</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="c3">c</font> )
 ;<br/><a NAME="E7:31_2_1_10"/>
<font color="Maroon" title="c6">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graph_4.html#E9:31"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E8:31_2_1_10"/><b>then </b>
<font color="Maroon" title="c32">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_10"><i><font color="Green" title="E63">A63</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E9:31_2_1_10"/>
<font color="Maroon" title="c30">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">j</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_10"><i><font color="Green" title="E63">A63</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_10"><i><font color="Green" title="E64">A64</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><a NAME="E79:31_2_1"/><b>then </b><i><font color="Green" title="E63">A65</font></i>: 
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">DL</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c29">DR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E75:31_2_1"><i><font color="Green" title="E62">A60</font></i></a>, <a class="ref" href="finseq_3.html#T48" title="FINSEQ_3:th.48">FINSEQ_3:48</a></i>;<br/><a NAME="E80:31_2_1"/>
<font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">lc21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E26:31_2_1"><i><font color="Green" title="E40">A22</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E64">A66</font></i>:  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c29">DR</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">c2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<b>by </b><i><a class="ref" href="graph_2.html#T4" title="GRAPH_2:th.4">GRAPH_2:4</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">c2</font>

;<br/><a NAME="E82:31_2_1"/><i><font color="Green" title="E65">A67</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c28">DL</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c29">DR</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E75:31_2_1"><i><font color="Green" title="E62">A60</font></i></a>, <a class="ref" href="finseq_3.html#T44" title="FINSEQ_3:th.44">FINSEQ_3:44</a></i>;<br/><a NAME="E83:31_2_1"/>
<font color="Maroon" title="c28">DL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D1" title="FINSEQ_1:def.1">FINSEQ_1:def 1</a></i>;<br/><a NAME="E84:31_2_1"/><b>then </b><i><font color="Green" title="E66">A68</font></i>: 
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c6">n1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T54" title="FINSEQ_3:th.54">FINSEQ_3:54</a></i>;<br/><a NAME="E85:31_2_1"/><b>then </b><i><font color="Green" title="E67">A69</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n1</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/><a NAME="E86:31_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">DL</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E75:31_2_1"><i><font color="Green" title="E62">A60</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E87:31_2_1"/><b>then </b><i><font color="Green" title="E68">A70</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">fc</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E39:31_2_1"><i><font color="Green" title="E48">A30</font></i></a>, <a class="txt" href="graph_4.html#E74:31_2_1"><i><font color="Green" title="E61">A59</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E88:31_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">DR</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E75:31_2_1"><i><font color="Green" title="E62">A60</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E89:31_2_1"/><b>then </b><i><font color="Green" title="E69">A71</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">fc</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E39:31_2_1"><i><font color="Green" title="E48">A30</font></i></a>, <a class="txt" href="graph_4.html#E74:31_2_1"><i><font color="Green" title="E61">A59</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>set </b><font color="Maroon" title="c30">SL</font> =  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font>;<br/><b>set </b><font color="Maroon" title="c31">SR</font> =  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_12"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c32">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_1_12_1"/><i><font color="Green" title="E70">A72</font></i>: 
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">c1</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c33">x</font>, <font color="Maroon" title="c34">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_2_1_12_1"/><i><font color="Green" title="E71">A73</font></i>: 
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c33">x</font>,<font color="Maroon" title="c34">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:31_2_1_12_1"/><i><font color="Green" title="E72">A74</font></i>: 
( <font color="Maroon" title="c33">x</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="c7">c1</font> &amp; <font color="Maroon" title="c34">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_1_12_1"><i><font color="Green" title="E70">A72</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_12_1"><i><font color="Green" title="E71">A73</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c35">k</font> = <font color="Maroon" title="c33">x</font> as    <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>  ;<br/><a NAME="E6:31_2_1_12_1"/><i><font color="Green" title="E73">A75</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">k</font> &amp; <font color="Maroon" title="c35">k</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="c7">c1</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_12_1"><i><font color="Green" title="E72">A74</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:31_2_1_12_1"/><b>then </b><i><font color="Green" title="E74">A76</font></i>: 
<font color="Maroon" title="c33">x</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E56:31_2_1"><i><font color="Green" title="E57">A51</font></i></a>, <a class="txt" href="graph_4.html#E85:31_2_1"><i><font color="Green" title="E67">A69</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:31_2_1_12_1"/><b>then </b><i><font color="Green" title="E75">A77</font></i>: 
<font color="Maroon" title="c35">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E84:31_2_1"><i><font color="Green" title="E66">A68</font></i></a>, <a class="txt" href="graph_4.html#E85:31_2_1"><i><font color="Green" title="E67">A69</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><a NAME="E9:31_2_1_12_1"/><i><font color="Green" title="E76">A78</font></i>: 
<font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) or ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c3">c</font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E56:31_2_1"><i><font color="Green" title="E57">A51</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_12_1"><i><font color="Green" title="E73">A75</font></i></a></i>;<br/><a NAME="E10:31_2_1_12_1"/><b>then </b><i><font color="Green" title="E77">A79</font></i>: 
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E39:31_2_1"><i><font color="Green" title="E48">A30</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_1_12_1"><i><font color="Green" title="E74">A76</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_1_12_1"><i><font color="Green" title="E75">A77</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E11:31_2_1_12_1"/>
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> <font color="Maroon" title="c35">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_12_1"><i><font color="Green" title="E72">A74</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c36">i</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> <b> such that </b><br/><a NAME="E13:31_2_1_12_1"/><i><font color="Green" title="E78">A80</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">i</font> &amp; <font color="Maroon" title="c36">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n1</font> &amp; <font color="Maroon" title="c35">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E56:31_2_1"><i><font color="Green" title="E57">A51</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_12_1"><i><font color="Green" title="E73">A75</font></i></a>, <a class="ref" href="graph_2.html#T1" title="GRAPH_2:th.1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c19">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> = 
<font color="Maroon" title="c19">fc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">k</font>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31_2_1_12_1"><i><font color="Green" title="E75">A77</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_12_1"><i><font color="Green" title="E77">A79</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c36">i</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E39:31_2_1"><i><font color="Green" title="E48">A30</font></i></a>, <a class="txt" href="graph_4.html#E43:31_2_1"><i><font color="Green" title="E50">A33</font></i></a>, <a class="txt" href="graph_4.html#E9:31_2_1_12_1"><i><font color="Green" title="E76">A78</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_12_1"><i><font color="Green" title="E78">A80</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c34">y</font>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_1"><i><font color="Green" title="E29">A11</font></i></a>, <a class="txt" href="graph_4.html#E56:31_2_1"><i><font color="Green" title="E57">A51</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_12_1"><i><font color="Green" title="E72">A74</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_12_1"><i><font color="Green" title="E78">A80</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green" title="E17">Lm2</font></i></a></i>
;<br/><b>hence </b><a NAME="E15:31_2_1_12_1"/>
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_12_1"><i><font color="Green" title="E71">A73</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_12_1"><i><font color="Green" title="E77">A79</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_1_12"/><i><font color="Green" title="E70">A81</font></i>: 
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c33">x</font>, <font color="Maroon" title="c34">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:31_2_1_12"/><i><font color="Green" title="E71">A82</font></i>: 
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c33">x</font>,<font color="Maroon" title="c34">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:31_2_1_12"/><i><font color="Green" title="E72">A83</font></i>: 
( <font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c34">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_1_12"><i><font color="Green" title="E70">A81</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_12"><i><font color="Green" title="E71">A82</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E6:31_2_1_12"/><b>then </b><i><font color="Green" title="E73">A84</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">fc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E7:31_2_1_12"/><i><font color="Green" title="E74">A85</font></i>: 
( <font color="Maroon" title="c33">x</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">fc</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E5:31_2_1_12"><i><font color="Green" title="E72">A83</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E8:31_2_1_12"/><b>then </b>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E85:31_2_1"><i><font color="Green" title="E67">A69</font></i></a>, <a class="ref" href="finseq_1.html#D1" title="FINSEQ_1:def.1">FINSEQ_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c35">k</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> <b> such that </b><br/><a NAME="E10:31_2_1_12"/><i><font color="Green" title="E75">A86</font></i>: 
( <font color="Maroon" title="c35">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">x</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">k</font> &amp; <font color="Maroon" title="c35">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> )
 ;<br/><a NAME="E11:31_2_1_12"/><i><font color="Green" title="E76">A87</font></i>: 
<font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">fc</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E39:31_2_1"><i><font color="Green" title="E48">A30</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_12"><i><font color="Green" title="E75">A86</font></i></a></i>;<br/><a NAME="E12:31_2_1_12"/><i><font color="Green" title="E77">A88</font></i>: 
<font color="Maroon" title="c35">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E84:31_2_1"><i><font color="Green" title="E66">A68</font></i></a>, <a class="txt" href="graph_4.html#E85:31_2_1"><i><font color="Green" title="E67">A69</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_1_12"><i><font color="Green" title="E74">A85</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_12"><i><font color="Green" title="E75">A86</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><a NAME="E13:31_2_1_12"/><i><font color="Green" title="E78">A89</font></i>: 
<font color="Maroon" title="c35">k</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="c7">c1</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E56:31_2_1"><i><font color="Green" title="E57">A51</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_12"><i><font color="Green" title="E75">A86</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:31_2_1_12"/>
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> <font color="Maroon" title="c35">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1_12"><i><font color="Green" title="E75">A86</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c36">i</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> <b> such that </b><br/><a NAME="E16:31_2_1_12"/><i><font color="Green" title="E79">A90</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">i</font> &amp; <font color="Maroon" title="c36">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n1</font> &amp; <font color="Maroon" title="c35">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1_12"><i><font color="Green" title="E75">A86</font></i></a>, <a class="ref" href="graph_2.html#T1" title="GRAPH_2:th.1">GRAPH_2:1</a></i>;<br/><font color="Maroon" title="c7">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">k</font> = 
<font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">k</font>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_1"><i><font color="Green" title="E29">A11</font></i></a>, <a class="txt" href="graph_4.html#E56:31_2_1"><i><font color="Green" title="E57">A51</font></i></a>, <a class="txt" href="graph_4.html#E16:31_2_1_12"><i><font color="Green" title="E79">A90</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green" title="E17">Lm2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c34">y</font>
<b>by </b><i><a class="txt" href="graph_4.html#E43:31_2_1"><i><font color="Green" title="E50">A33</font></i></a>, <a class="txt" href="graph_4.html#E5:31_2_1_12"><i><font color="Green" title="E72">A83</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_12"><i><font color="Green" title="E73">A84</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_12"><i><font color="Green" title="E75">A86</font></i></a>, <a class="txt" href="graph_4.html#E11:31_2_1_12"><i><font color="Green" title="E76">A87</font></i></a>, <a class="txt" href="graph_4.html#E12:31_2_1_12"><i><font color="Green" title="E77">A88</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
;<br/><b>hence </b><a NAME="E18:31_2_1_12"/>
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">c1</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_12"><i><font color="Green" title="E71">A82</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_12"><i><font color="Green" title="E75">A86</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_12"><i><font color="Green" title="E78">A89</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E91:31_2_1"/><b>then </b><i><font color="Green" title="E70">A91</font></i>: 
<font color="Maroon" title="c7">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_13"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c32">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_1_13_1"/><i><font color="Green" title="E71">A92</font></i>: 
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">c2</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c33">x</font>, <font color="Maroon" title="c34">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_2_1_13_1"/><i><font color="Green" title="E72">A93</font></i>: 
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c33">x</font>,<font color="Maroon" title="c34">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:31_2_1_13_1"/><i><font color="Green" title="E73">A94</font></i>: 
( <font color="Maroon" title="c33">x</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="c8">c2</font> &amp; <font color="Maroon" title="c34">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_1_13_1"><i><font color="Green" title="E71">A92</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_13_1"><i><font color="Green" title="E72">A93</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c35">k</font> = <font color="Maroon" title="c33">x</font> as    <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>  ;<br/><a NAME="E6:31_2_1_13_1"/><i><font color="Green" title="E74">A95</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">k</font> &amp; <font color="Maroon" title="c35">k</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="c8">c2</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_13_1"><i><font color="Green" title="E73">A94</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:31_2_1_13_1"/><i><font color="Green" title="E75">A96</font></i>: 
<font color="Maroon" title="c33">x</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E81:31_2_1"><i><font color="Green" title="E64">A66</font></i></a>, <a class="txt" href="graph_4.html#E82:31_2_1"><i><font color="Green" title="E65">A67</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_13_1"><i><font color="Green" title="E73">A94</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E8:31_2_1_13_1"/><i><font color="Green" title="E76">A97</font></i>: 
<font color="Maroon" title="c14">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c35">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E24:31_2_1"><i><font color="Green" title="E39">A21</font></i></a>, <a class="txt" href="graph_4.html#E26:31_2_1"><i><font color="Green" title="E40">A22</font></i></a>, <a class="txt" href="graph_4.html#E30:31_2_1"><i><font color="Green" title="E42">A24</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_13_1"><i><font color="Green" title="E74">A95</font></i></a>, <a class="ref" href="graph_2.html#T5" title="GRAPH_2:th.5">GRAPH_2:5</a></i>;<br/><a NAME="E9:31_2_1_13_1"/><i><font color="Green" title="E77">A98</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E7:31_2_1_13_1"><i><font color="Green" title="E75">A96</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:31_2_1_13_1"/><b>then </b><i><font color="Green" title="E78">A99</font></i>: 
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E89:31_2_1"><i><font color="Green" title="E69">A71</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_1_13_1"><i><font color="Green" title="E75">A96</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E11:31_2_1_13_1"/>
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> <font color="Maroon" title="c35">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_13_1"><i><font color="Green" title="E73">A94</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c36">i</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> <b> such that </b><br/><a NAME="E13:31_2_1_13_1"/><i><font color="Green" title="E79">A100</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">i</font> &amp; <font color="Maroon" title="c36">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> <font color="Maroon" title="c8">c2</font> &amp; <font color="Maroon" title="c35">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_1_13_1"><i><font color="Green" title="E74">A95</font></i></a>, <a class="ref" href="graph_2.html#T1" title="GRAPH_2:th.1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c19">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> = 
<font color="Maroon" title="c19">fc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c35">k</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31_2_1_13_1"><i><font color="Green" title="E76">A97</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_13_1"><i><font color="Green" title="E78">A99</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c36">i</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E43:31_2_1"><i><font color="Green" title="E50">A33</font></i></a>, <a class="txt" href="graph_4.html#E89:31_2_1"><i><font color="Green" title="E69">A71</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_1_13_1"><i><font color="Green" title="E76">A97</font></i></a>, <a class="txt" href="graph_4.html#E9:31_2_1_13_1"><i><font color="Green" title="E77">A98</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a>, <a class="txt" href="graph_4.html#E13:31_2_1_13_1"><i><font color="Green" title="E79">A100</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c34">y</font>
<b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_13_1"><i><font color="Green" title="E73">A94</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_13_1"><i><font color="Green" title="E79">A100</font></i></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></i>
;<br/><b>hence </b><a NAME="E15:31_2_1_13_1"/>
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_13_1"><i><font color="Green" title="E72">A93</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_13_1"><i><font color="Green" title="E78">A99</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_1_13"/><i><font color="Green" title="E71">A101</font></i>: 
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c33">x</font>, <font color="Maroon" title="c34">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:31_2_1_13"/><i><font color="Green" title="E72">A102</font></i>: 
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c33">x</font>,<font color="Maroon" title="c34">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:31_2_1_13"/><i><font color="Green" title="E73">A103</font></i>: 
( <font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c34">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_1_13"><i><font color="Green" title="E71">A101</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_13"><i><font color="Green" title="E72">A102</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E6:31_2_1_13"/><b>then </b><i><font color="Green" title="E74">A104</font></i>: 
( <font color="Maroon" title="c33">x</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">fc</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c35">k</font> = <font color="Maroon" title="c33">x</font> as    <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>  ;<br/><a NAME="E8:31_2_1_13"/><i><font color="Green" title="E75">A105</font></i>: 
<font color="Maroon" title="c35">k</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="c8">c2</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E81:31_2_1"><i><font color="Green" title="E64">A66</font></i></a>, <a class="txt" href="graph_4.html#E82:31_2_1"><i><font color="Green" title="E65">A67</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_13"><i><font color="Green" title="E74">A104</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E9:31_2_1_13"/><i><font color="Green" title="E76">A106</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">k</font> &amp; <font color="Maroon" title="c35">k</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="c8">c2</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E81:31_2_1"><i><font color="Green" title="E64">A66</font></i></a>, <a class="txt" href="graph_4.html#E82:31_2_1"><i><font color="Green" title="E65">A67</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_13"><i><font color="Green" title="E74">A104</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:31_2_1_13"/><b>then </b><i><font color="Green" title="E77">A107</font></i>: 
<font color="Maroon" title="c14">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c35">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E24:31_2_1"><i><font color="Green" title="E39">A21</font></i></a>, <a class="txt" href="graph_4.html#E26:31_2_1"><i><font color="Green" title="E40">A22</font></i></a>, <a class="txt" href="graph_4.html#E30:31_2_1"><i><font color="Green" title="E42">A24</font></i></a>, <a class="ref" href="graph_2.html#T5" title="GRAPH_2:th.5">GRAPH_2:5</a></i>;<br/><a NAME="E11:31_2_1_13"/>
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> <font color="Maroon" title="c35">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_1_13"><i><font color="Green" title="E74">A104</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c36">i</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> <b> such that </b><br/><a NAME="E13:31_2_1_13"/><i><font color="Green" title="E78">A108</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">i</font> &amp; <font color="Maroon" title="c36">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> <font color="Maroon" title="c8">c2</font> &amp; <font color="Maroon" title="c35">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_1_13"><i><font color="Green" title="E76">A106</font></i></a>, <a class="ref" href="graph_2.html#T1" title="GRAPH_2:th.1">GRAPH_2:1</a></i>;<br/><font color="Maroon" title="c8">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">k</font> = 
<font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c36">i</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_13"><i><font color="Green" title="E78">A108</font></i></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c19">fc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">k</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_1_13"><i><font color="Green" title="E78">A108</font></i></a>, <a class="txt" href="graph_4.html#E43:31_2_1"><i><font color="Green" title="E50">A33</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_13"><i><font color="Green" title="E74">A104</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_13"><i><font color="Green" title="E77">A107</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c34">y</font>
<b>by </b><i><a class="txt" href="graph_4.html#E5:31_2_1_13"><i><font color="Green" title="E73">A103</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_13"><i><font color="Green" title="E74">A104</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E15:31_2_1_13"/>
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">c2</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_13"><i><font color="Green" title="E72">A102</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_1_13"><i><font color="Green" title="E75">A105</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E93:31_2_1"/><b>then </b><i><font color="Green" title="E71">A109</font></i>: 
<font color="Maroon" title="c8">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E94:31_2_1"/>
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c19">fc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">DL</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">DR</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E39:31_2_1"><i><font color="Green" title="E48">A30</font></i></a>, <a class="txt" href="graph_4.html#E74:31_2_1"><i><font color="Green" title="E61">A59</font></i></a>, <a class="txt" href="graph_4.html#E79:31_2_1"><i><font color="Green" title="E63">A65</font></i></a>, <a class="ref" href="finseq_1.html#D14" title="FINSEQ_1:def.14">FINSEQ_1:def 14</a></i>;<br/><b>hence </b><a NAME="E95:31_2_1"/>
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c19">fc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">c'</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E87:31_2_1"><i><font color="Green" title="E68">A70</font></i></a>, <a class="txt" href="graph_4.html#E89:31_2_1"><i><font color="Green" title="E69">A71</font></i></a>, <a class="txt" href="graph_4.html#E91:31_2_1"><i><font color="Green" title="E70">A91</font></i></a>, <a class="txt" href="graph_4.html#E93:31_2_1"><i><font color="Green" title="E71">A109</font></i></a>, <a class="ref" href="graph_2.html#T26" title="GRAPH_2:th.26">GRAPH_2:26</a></i>;<br/><b>set </b><font color="Maroon" title="c32">DL</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c33">DR</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</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="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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">vs</font> ) </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_14"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c34">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_1_14_1"/>
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) or ( <font color="Maroon" title="c5">m</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="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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">vs</font> ) ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c35">k</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> <b> such that </b><br/><a NAME="E3:31_2_1_14_1"/><i><font color="Green" title="E72">A110</font></i>: 
( <font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">k</font> &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">k</font> &amp; <font color="Maroon" title="c35">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) or ( <font color="Maroon" title="c5">m</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="c35">k</font> &amp; <font color="Maroon" title="c35">k</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">vs</font> ) ) )
 ;<br/><a NAME="E4:31_2_1_14_1"/>
( <font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span>  or <font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</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="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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">vs</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_14_1"><i><font color="Green" title="E72">A110</font></i></a></i>;<br/><b>hence </b><a NAME="E5:31_2_1_14_1"/>
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</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="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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">vs</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_1_14"/><i><font color="Green" title="E72">A111</font></i>: 
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</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="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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">vs</font> ) </span>}</span> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_14_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_14_2"/>
( <font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span>  or <font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</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="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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">vs</font> ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_1_14"><i><font color="Green" title="E72">A111</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_14_2_1"><b>suppose </b></a><a NAME="E1:31_2_1_14_2_1"/>
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c35">k</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> <b> such that </b><br/><a NAME="E3:31_2_1_14_2_1"/><i><font color="Green" title="E73">A112</font></i>: 
( <font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">k</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">k</font> &amp; <font color="Maroon" title="c35">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> )
 ;<br/><b>thus </b><a NAME="E4:31_2_1_14_2_1"/>
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) or ( <font color="Maroon" title="c5">m</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="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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">vs</font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_14_2_1"><i><font color="Green" title="E73">A112</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_14_2_2"><b>suppose </b></a><a NAME="E1:31_2_1_14_2_2"/>
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</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="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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">vs</font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c35">k</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> <b> such that </b><br/><a NAME="E3:31_2_1_14_2_2"/><i><font color="Green" title="E73">A113</font></i>: 
( <font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">k</font> &amp; <font color="Maroon" title="c5">m</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="c35">k</font> &amp; <font color="Maroon" title="c35">k</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">vs</font> )
 ;<br/><b>thus </b><a NAME="E4:31_2_1_14_2_2"/>
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) or ( <font color="Maroon" title="c5">m</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="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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">vs</font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_14_2_2"><i><font color="Green" title="E73">A113</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E97:31_2_1"/><b>then </b><i><font color="Green" title="E72">A114</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) or ( <font color="Maroon" title="c5">m</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="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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">vs</font> ) ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</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="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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">vs</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E98:31_2_1"/><i><font color="Green" title="E73">A115</font></i>: 
( <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span> &amp; <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</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="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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">vs</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_15"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>
<div class="add"><b>let </b><font color="Maroon" title="c34">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:31_2_1_15_1"/>
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c35">k</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> <b> such that </b><br/><a NAME="E3:31_2_1_15_1"/><i><font color="Green" title="E74">A116</font></i>: 
( <font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">k</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">k</font> &amp; <font color="Maroon" title="c35">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> )
 ;<br/><a NAME="E4:31_2_1_15_1"/>
<font color="Maroon" title="c35">k</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">vs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E15:31_2_1"><i><font color="Green" title="E33">A15</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_15_1"><i><font color="Green" title="E74">A116</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:31_2_1_15_1"/>
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_15_1"><i><font color="Green" title="E74">A116</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c34">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:31_2_1_15"/>
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</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="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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">vs</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c35">k</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> <b> such that </b><br/><a NAME="E4:31_2_1_15"/><i><font color="Green" title="E74">A117</font></i>: 
( <font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">k</font> &amp; <font color="Maroon" title="c5">m</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="c35">k</font> &amp; <font color="Maroon" title="c35">k</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">vs</font> )
 ;<br/>
<a NAME="E5:31_2_1_15"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E6:31_2_1_15"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">k</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_15"><i><font color="Green" title="E74">A117</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E7:31_2_1_15"/>
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_15"><i><font color="Green" title="E74">A117</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c34">DL</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c35">DR</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( <font color="Maroon" title="c5">m</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="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</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">vs</font> ) </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="graph_4.html#E98:31_2_1"><i><font color="Green" title="E73">A115</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_16"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c36">i</font>, <font color="Maroon" title="c37">j</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> ;<br/><b>assume </b><a NAME="E1:31_2_1_16"/>
<font color="Maroon" title="c36">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">DL</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c38">k</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> <b> such that </b><br/><a NAME="E3:31_2_1_16"/><i><font color="Green" title="E74">A118</font></i>: 
( <font color="Maroon" title="c38">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">k</font> &amp; <font color="Maroon" title="c38">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> )
 ;<br/><b>assume </b><a NAME="E4:31_2_1_16"/>
<font color="Maroon" title="c37">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">DR</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c39">l</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> <b> such that </b><br/><a NAME="E6:31_2_1_16"/><i><font color="Green" title="E75">A119</font></i>: 
( <font color="Maroon" title="c39">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">j</font> &amp; <font color="Maroon" title="c5">m</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="c39">l</font> &amp; <font color="Maroon" title="c39">l</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">vs</font> )
 ;<br/><a NAME="E7:31_2_1_16"/>
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E8:31_2_1_16"/><b>then </b><i><font color="Green" title="E76">A120</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">l</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_1_16"><i><font color="Green" title="E75">A119</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:31_2_1_16"/>
<font color="Maroon" title="c38">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_16"><i><font color="Green" title="E74">A118</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E10:31_2_1_16"/>
<font color="Maroon" title="c36">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c37">j</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_16"><i><font color="Green" title="E74">A118</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_16"><i><font color="Green" title="E75">A119</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_1_16"><i><font color="Green" title="E76">A120</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><a NAME="E102:31_2_1"/><b>then </b><i><font color="Green" title="E74">A121</font></i>: 
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">DL</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c35">DR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E98:31_2_1"><i><font color="Green" title="E73">A115</font></i></a>, <a class="ref" href="finseq_3.html#T48" title="FINSEQ_3:th.48">FINSEQ_3:48</a></i>;<br/><a NAME="E103:31_2_1"/>
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> <font color="Maroon" title="c13">p2</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_1"><i><font color="Green" title="E38">A20</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E104:31_2_1"/><b>then </b>
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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">p2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E105:31_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">p2</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">p2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c36">lp21</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">p2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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>  ;<br/><a NAME="E107:31_2_1"/>
<font color="Maroon" title="c36">lp21</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">lp21</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_1"><i><font color="Green" title="E38">A20</font></i></a>, <a class="txt" href="graph_4.html#E26:31_2_1"><i><font color="Green" title="E40">A22</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_1"><i><font color="Green" title="E41">A23</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c37">lp22</font> = <font color="Maroon" title="c36">lp21</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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>  ;<br/><i><font color="Green" title="E75">A122</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c37">lp22</font> = 
<font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c36">lp21</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>

<br/>.= 
<font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E18:31_2_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="graph_4.html#E26:31_2_1"><i><font color="Green" title="E40">A22</font></i></a>, <a class="ref" href="graph_4.html#D5" target="_self" title="GRAPH_4:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1

;<br/><b>then </b><i><font color="Green" title="E76">A123</font></i>:  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c35">DR</font> = 
<font color="Maroon" title="c37">lp22</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="graph_2.html#T4" title="GRAPH_2:th.4">GRAPH_2:4</a></i>
<br/>.= 
<font color="Maroon" title="c36">lp21</font>

;<br/><a NAME="E111:31_2_1"/><i><font color="Green" title="E77">A124</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c5">m</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">p2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></i>
<br/>.= 
<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"><span class="p3">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="txt" href="graph_4.html#E111:31_2_1"><i><font color="Green" title="E77">A124</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green" title="E17">Lm2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">m</font>

;<br/><a NAME="E113:31_2_1"/><b>then </b><i><font color="Green" title="E78">A125</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">p2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span></span>)</span> <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> 1
 ;<br/><a NAME="E114:31_2_1"/><i><font color="Green" title="E79">A126</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c34">DL</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c35">DR</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E98:31_2_1"><i><font color="Green" title="E73">A115</font></i></a>, <a class="ref" href="finseq_3.html#T44" title="FINSEQ_3:th.44">FINSEQ_3:44</a></i>;<br/><a NAME="E115:31_2_1"/><i><font color="Green" title="E80">A127</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">pp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E19:31_2_1"><i><font color="Green" title="E37">A19</font></i></a>, <a class="txt" href="graph_4.html#E55:31_2_1"><i><font color="Green" title="E56">A50</font></i></a></i>;<br/><a NAME="E116:31_2_1"/>
<font color="Maroon" title="c34">DL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D1" title="FINSEQ_1:def.1">FINSEQ_1:def 1</a></i>;<br/><a NAME="E117:31_2_1"/><b>then </b><i><font color="Green" title="E81">A128</font></i>: 
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T54" title="FINSEQ_3:th.54">FINSEQ_3:54</a></i>;<br/><a NAME="E118:31_2_1"/><b>then </b><i><font color="Green" title="E82">A129</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/><a NAME="E119:31_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">DL</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E98:31_2_1"><i><font color="Green" title="E73">A115</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E120:31_2_1"/><b>then </b><i><font color="Green" title="E83">A130</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">fvs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_1"><i><font color="Green" title="E51">A39</font></i></a>, <a class="txt" href="graph_4.html#E97:31_2_1"><i><font color="Green" title="E72">A114</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E121:31_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">DR</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E98:31_2_1"><i><font color="Green" title="E73">A115</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E122:31_2_1"/><b>then </b><i><font color="Green" title="E84">A131</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">fvs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_1"><i><font color="Green" title="E51">A39</font></i></a>, <a class="txt" href="graph_4.html#E97:31_2_1"><i><font color="Green" title="E72">A114</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>set </b><font color="Maroon" title="c38">SL</font> =  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font>;<br/><b>set </b><font color="Maroon" title="c39">SR</font> =  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_20"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c40">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_1_20_1"/><i><font color="Green" title="E85">A132</font></i>: 
<font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">pp</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c41">x</font>, <font color="Maroon" title="c42">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_2_1_20_1"/><i><font color="Green" title="E86">A133</font></i>: 
<font color="Maroon" title="c40">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c41">x</font>,<font color="Maroon" title="c42">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:31_2_1_20_1"/><i><font color="Green" title="E87">A134</font></i>: 
( <font color="Maroon" title="c41">x</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="c12">pp</font> &amp; <font color="Maroon" title="c42">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_1_20_1"><i><font color="Green" title="E85">A132</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_20_1"><i><font color="Green" title="E86">A133</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c43">k</font> = <font color="Maroon" title="c41">x</font> as    <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>  ;<br/><a NAME="E6:31_2_1_20_1"/><i><font color="Green" title="E88">A135</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">k</font> &amp; <font color="Maroon" title="c43">k</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="c12">pp</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_20_1"><i><font color="Green" title="E87">A134</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:31_2_1_20_1"/><b>then </b><i><font color="Green" title="E89">A136</font></i>: 
<font color="Maroon" title="c41">x</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E115:31_2_1"><i><font color="Green" title="E80">A127</font></i></a>, <a class="txt" href="graph_4.html#E118:31_2_1"><i><font color="Green" title="E82">A129</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:31_2_1_20_1"/><b>then </b><i><font color="Green" title="E90">A137</font></i>: 
<font color="Maroon" title="c43">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E117:31_2_1"><i><font color="Green" title="E81">A128</font></i></a>, <a class="txt" href="graph_4.html#E118:31_2_1"><i><font color="Green" title="E82">A129</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><a NAME="E9:31_2_1_20_1"/><i><font color="Green" title="E91">A138</font></i>: 
<font color="Maroon" title="c43">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) or ( <font color="Maroon" title="c5">m</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="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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">vs</font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E115:31_2_1"><i><font color="Green" title="E80">A127</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_20_1"><i><font color="Green" title="E88">A135</font></i></a></i>;<br/><a NAME="E10:31_2_1_20_1"/><b>then </b><i><font color="Green" title="E92">A139</font></i>: 
<font color="Maroon" title="c41">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_1"><i><font color="Green" title="E51">A39</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_1_20_1"><i><font color="Green" title="E89">A136</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_1_20_1"><i><font color="Green" title="E90">A137</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E11:31_2_1_20_1"/>
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> <font color="Maroon" title="c43">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_20_1"><i><font color="Green" title="E87">A134</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c44">i</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> <b> such that </b><br/><a NAME="E13:31_2_1_20_1"/><i><font color="Green" title="E93">A140</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">i</font> &amp; <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c43">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E115:31_2_1"><i><font color="Green" title="E80">A127</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_20_1"><i><font color="Green" title="E88">A135</font></i></a>, <a class="ref" href="graph_2.html#T1" title="GRAPH_2:th.1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x</font> = 
<font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">k</font>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31_2_1_20_1"><i><font color="Green" title="E90">A137</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_20_1"><i><font color="Green" title="E92">A139</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c44">i</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_1"><i><font color="Green" title="E51">A39</font></i></a>, <a class="txt" href="graph_4.html#E50:31_2_1"><i><font color="Green" title="E53">A42</font></i></a>, <a class="txt" href="graph_4.html#E9:31_2_1_20_1"><i><font color="Green" title="E91">A138</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_20_1"><i><font color="Green" title="E93">A140</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c42">y</font>
<b>by </b><i><a class="txt" href="graph_4.html#E15:31_2_1"><i><font color="Green" title="E33">A15</font></i></a>, <a class="txt" href="graph_4.html#E115:31_2_1"><i><font color="Green" title="E80">A127</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_20_1"><i><font color="Green" title="E87">A134</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_20_1"><i><font color="Green" title="E93">A140</font></i></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></i>
;<br/><b>hence </b><a NAME="E15:31_2_1_20_1"/>
<font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_20_1"><i><font color="Green" title="E86">A133</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_20_1"><i><font color="Green" title="E92">A139</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_1_20"/><i><font color="Green" title="E85">A141</font></i>: 
<font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c41">x</font>, <font color="Maroon" title="c42">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:31_2_1_20"/><i><font color="Green" title="E86">A142</font></i>: 
<font color="Maroon" title="c40">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c41">x</font>,<font color="Maroon" title="c42">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:31_2_1_20"/><i><font color="Green" title="E87">A143</font></i>: 
( <font color="Maroon" title="c41">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c42">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_1_20"><i><font color="Green" title="E85">A141</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_20"><i><font color="Green" title="E86">A142</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E6:31_2_1_20"/><b>then </b><i><font color="Green" title="E88">A144</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E7:31_2_1_20"/><i><font color="Green" title="E89">A145</font></i>: 
( <font color="Maroon" title="c41">x</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">fvs</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E5:31_2_1_20"><i><font color="Green" title="E87">A143</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E8:31_2_1_20"/><b>then </b>
<font color="Maroon" title="c41">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E118:31_2_1"><i><font color="Green" title="E82">A129</font></i></a>, <a class="ref" href="finseq_1.html#D1" title="FINSEQ_1:def.1">FINSEQ_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c43">k</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> <b> such that </b><br/><a NAME="E10:31_2_1_20"/><i><font color="Green" title="E90">A146</font></i>: 
( <font color="Maroon" title="c43">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">x</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">k</font> &amp; <font color="Maroon" title="c43">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> )
 ;<br/><a NAME="E11:31_2_1_20"/><i><font color="Green" title="E91">A147</font></i>: 
<font color="Maroon" title="c43">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">fvs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_1"><i><font color="Green" title="E51">A39</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_20"><i><font color="Green" title="E90">A146</font></i></a></i>;<br/><a NAME="E12:31_2_1_20"/><i><font color="Green" title="E92">A148</font></i>: 
<font color="Maroon" title="c43">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E117:31_2_1"><i><font color="Green" title="E81">A128</font></i></a>, <a class="txt" href="graph_4.html#E118:31_2_1"><i><font color="Green" title="E82">A129</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_1_20"><i><font color="Green" title="E89">A145</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_20"><i><font color="Green" title="E90">A146</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><a NAME="E13:31_2_1_20"/><i><font color="Green" title="E93">A149</font></i>: 
<font color="Maroon" title="c43">k</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="c12">pp</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E115:31_2_1"><i><font color="Green" title="E80">A127</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_20"><i><font color="Green" title="E90">A146</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:31_2_1_20"/>
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> <font color="Maroon" title="c43">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1_20"><i><font color="Green" title="E90">A146</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c44">i</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> <b> such that </b><br/><a NAME="E16:31_2_1_20"/><i><font color="Green" title="E94">A150</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">i</font> &amp; <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c43">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1_20"><i><font color="Green" title="E90">A146</font></i></a>, <a class="ref" href="graph_2.html#T1" title="GRAPH_2:th.1">GRAPH_2:1</a></i>;<br/><font color="Maroon" title="c12">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">k</font> = 
<font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">k</font>
<b>by </b><i><a class="txt" href="graph_4.html#E15:31_2_1"><i><font color="Green" title="E33">A15</font></i></a>, <a class="txt" href="graph_4.html#E115:31_2_1"><i><font color="Green" title="E80">A127</font></i></a>, <a class="txt" href="graph_4.html#E16:31_2_1_20"><i><font color="Green" title="E94">A150</font></i></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c42">y</font>
<b>by </b><i><a class="txt" href="graph_4.html#E50:31_2_1"><i><font color="Green" title="E53">A42</font></i></a>, <a class="txt" href="graph_4.html#E5:31_2_1_20"><i><font color="Green" title="E87">A143</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_20"><i><font color="Green" title="E88">A144</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_20"><i><font color="Green" title="E90">A146</font></i></a>, <a class="txt" href="graph_4.html#E11:31_2_1_20"><i><font color="Green" title="E91">A147</font></i></a>, <a class="txt" href="graph_4.html#E12:31_2_1_20"><i><font color="Green" title="E92">A148</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
;<br/><b>hence </b><a NAME="E18:31_2_1_20"/>
<font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">pp</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_20"><i><font color="Green" title="E86">A142</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_20"><i><font color="Green" title="E90">A146</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_20"><i><font color="Green" title="E93">A149</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E124:31_2_1"/><b>then </b><i><font color="Green" title="E85">A151</font></i>: 
<font color="Maroon" title="c12">pp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E125:31_2_1"/><i><font color="Green" title="E86">A152</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c37">lp22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c36">lp21</font>
 ;<br/><a NAME="E126:31_2_1"/><i><font color="Green" title="E87">A153</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c5">m</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_21"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c40">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_1_21_1"/><i><font color="Green" title="E88">A154</font></i>: 
<font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c41">x</font>, <font color="Maroon" title="c42">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_2_1_21_1"/><i><font color="Green" title="E89">A155</font></i>: 
<font color="Maroon" title="c40">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c41">x</font>,<font color="Maroon" title="c42">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:31_2_1_21_1"/><i><font color="Green" title="E90">A156</font></i>: 
( <font color="Maroon" title="c41">x</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span> &amp; <font color="Maroon" title="c42">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_1_21_1"><i><font color="Green" title="E88">A154</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_21_1"><i><font color="Green" title="E89">A155</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c43">k</font> = <font color="Maroon" title="c41">x</font> as    <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>  ;<br/><a NAME="E6:31_2_1_21_1"/><i><font color="Green" title="E91">A157</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">k</font> &amp; <font color="Maroon" title="c43">k</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_21_1"><i><font color="Green" title="E90">A156</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:31_2_1_21_1"/><i><font color="Green" title="E92">A158</font></i>: 
<font color="Maroon" title="c41">x</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E110:31_2_1"><i><font color="Green" title="E76">A123</font></i></a>, <a class="txt" href="graph_4.html#E113:31_2_1"><i><font color="Green" title="E78">A125</font></i></a>, <a class="txt" href="graph_4.html#E114:31_2_1"><i><font color="Green" title="E79">A126</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_21_1"><i><font color="Green" title="E90">A156</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E8:31_2_1_21_1"/><i><font color="Green" title="E93">A159</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c43">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="graph_4.html#E109:31_2_1"><i><font color="Green" title="E75">A122</font></i></a>, <a class="txt" href="graph_4.html#E113:31_2_1"><i><font color="Green" title="E78">A125</font></i></a>, <a class="txt" href="graph_4.html#E125:31_2_1"><i><font color="Green" title="E86">A152</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_21_1"><i><font color="Green" title="E91">A157</font></i></a>, <a class="ref" href="graph_2.html#T5" title="GRAPH_2:th.5">GRAPH_2:5</a></i>;<br/><a NAME="E9:31_2_1_21_1"/><i><font color="Green" title="E94">A160</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E7:31_2_1_21_1"><i><font color="Green" title="E92">A158</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:31_2_1_21_1"/><b>then </b><i><font color="Green" title="E95">A161</font></i>: 
<font color="Maroon" title="c41">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E122:31_2_1"><i><font color="Green" title="E84">A131</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_1_21_1"><i><font color="Green" title="E92">A158</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E11:31_2_1_21_1"/>
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> <font color="Maroon" title="c43">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_21_1"><i><font color="Green" title="E90">A156</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c44">i</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> <b> such that </b><br/><a NAME="E13:31_2_1_21_1"/><i><font color="Green" title="E96">A162</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">i</font> &amp; <font color="Maroon" title="c44">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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span> &amp; <font color="Maroon" title="c43">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_1_21_1"><i><font color="Green" title="E91">A157</font></i></a>, <a class="ref" href="graph_2.html#T1" title="GRAPH_2:th.1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x</font> = 
<font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c43">k</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31_2_1_21_1"><i><font color="Green" title="E93">A159</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_21_1"><i><font color="Green" title="E95">A161</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c44">i</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E50:31_2_1"><i><font color="Green" title="E53">A42</font></i></a>, <a class="txt" href="graph_4.html#E122:31_2_1"><i><font color="Green" title="E84">A131</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_1_21_1"><i><font color="Green" title="E93">A159</font></i></a>, <a class="txt" href="graph_4.html#E9:31_2_1_21_1"><i><font color="Green" title="E94">A160</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a>, <a class="txt" href="graph_4.html#E13:31_2_1_21_1"><i><font color="Green" title="E96">A162</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c42">y</font>
<b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="graph_4.html#E126:31_2_1"><i><font color="Green" title="E87">A153</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_21_1"><i><font color="Green" title="E90">A156</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_21_1"><i><font color="Green" title="E96">A162</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green" title="E17">Lm2</font></i></a></i>
;<br/><b>hence </b><a NAME="E15:31_2_1_21_1"/>
<font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_21_1"><i><font color="Green" title="E89">A155</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_21_1"><i><font color="Green" title="E95">A161</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_1_21"/><i><font color="Green" title="E88">A163</font></i>: 
<font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c41">x</font>, <font color="Maroon" title="c42">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:31_2_1_21"/><i><font color="Green" title="E89">A164</font></i>: 
<font color="Maroon" title="c40">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c41">x</font>,<font color="Maroon" title="c42">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:31_2_1_21"/><i><font color="Green" title="E90">A165</font></i>: 
( <font color="Maroon" title="c41">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c42">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_1_21"><i><font color="Green" title="E88">A163</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_21"><i><font color="Green" title="E89">A164</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E6:31_2_1_21"/><b>then </b><i><font color="Green" title="E91">A166</font></i>: 
( <font color="Maroon" title="c41">x</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">fvs</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c43">k</font> = <font color="Maroon" title="c41">x</font> as    <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>  ;<br/><a NAME="E8:31_2_1_21"/><i><font color="Green" title="E92">A167</font></i>: 
<font color="Maroon" title="c43">k</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E110:31_2_1"><i><font color="Green" title="E76">A123</font></i></a>, <a class="txt" href="graph_4.html#E113:31_2_1"><i><font color="Green" title="E78">A125</font></i></a>, <a class="txt" href="graph_4.html#E114:31_2_1"><i><font color="Green" title="E79">A126</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_21"><i><font color="Green" title="E91">A166</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E9:31_2_1_21"/><i><font color="Green" title="E93">A168</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">k</font> &amp; <font color="Maroon" title="c43">k</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_4.html#E110:31_2_1"><i><font color="Green" title="E76">A123</font></i></a>, <a class="txt" href="graph_4.html#E113:31_2_1"><i><font color="Green" title="E78">A125</font></i></a>, <a class="txt" href="graph_4.html#E114:31_2_1"><i><font color="Green" title="E79">A126</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_21"><i><font color="Green" title="E91">A166</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:31_2_1_21"/><b>then </b><i><font color="Green" title="E94">A169</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c43">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="graph_4.html#E109:31_2_1"><i><font color="Green" title="E75">A122</font></i></a>, <a class="txt" href="graph_4.html#E113:31_2_1"><i><font color="Green" title="E78">A125</font></i></a>, <a class="txt" href="graph_4.html#E125:31_2_1"><i><font color="Green" title="E86">A152</font></i></a>, <a class="ref" href="graph_2.html#T5" title="GRAPH_2:th.5">GRAPH_2:5</a></i>;<br/><a NAME="E11:31_2_1_21"/>
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> <font color="Maroon" title="c43">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_1_21"><i><font color="Green" title="E91">A166</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c44">i</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> <b> such that </b><br/><a NAME="E13:31_2_1_21"/><i><font color="Green" title="E95">A170</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">i</font> &amp; <font color="Maroon" title="c44">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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span> &amp; <font color="Maroon" title="c43">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_1_21"><i><font color="Green" title="E93">A168</font></i></a>, <a class="ref" href="graph_2.html#T1" title="GRAPH_2:th.1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">k</font> = 
<font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c44">i</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="graph_4.html#E126:31_2_1"><i><font color="Green" title="E87">A153</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_21"><i><font color="Green" title="E95">A170</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green" title="E17">Lm2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">k</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_1_21"><i><font color="Green" title="E95">A170</font></i></a>, <a class="txt" href="graph_4.html#E50:31_2_1"><i><font color="Green" title="E53">A42</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_21"><i><font color="Green" title="E91">A166</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_21"><i><font color="Green" title="E94">A169</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c42">y</font>
<b>by </b><i><a class="txt" href="graph_4.html#E5:31_2_1_21"><i><font color="Green" title="E90">A165</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_21"><i><font color="Green" title="E91">A166</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E15:31_2_1_21"/>
<font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_21"><i><font color="Green" title="E89">A164</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_1_21"><i><font color="Green" title="E92">A167</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E128:31_2_1"/><b>then </b><i><font color="Green" title="E88">A171</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E129:31_2_1"/>
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c23">fvs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c34">DL</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c35">DR</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_1"><i><font color="Green" title="E51">A39</font></i></a>, <a class="txt" href="graph_4.html#E97:31_2_1"><i><font color="Green" title="E72">A114</font></i></a>, <a class="txt" href="graph_4.html#E102:31_2_1"><i><font color="Green" title="E74">A121</font></i></a>, <a class="ref" href="finseq_1.html#D14" title="FINSEQ_1:def.14">FINSEQ_1:def 14</a></i>;<br/><b>hence </b><a NAME="E130:31_2_1"/>
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c23">fvs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">p1</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E54:31_2_1"><i><font color="Green" title="E55">A49</font></i></a>, <a class="txt" href="graph_4.html#E120:31_2_1"><i><font color="Green" title="E83">A130</font></i></a>, <a class="txt" href="graph_4.html#E122:31_2_1"><i><font color="Green" title="E84">A131</font></i></a>, <a class="txt" href="graph_4.html#E124:31_2_1"><i><font color="Green" title="E85">A151</font></i></a>, <a class="txt" href="graph_4.html#E128:31_2_1"><i><font color="Green" title="E88">A171</font></i></a>, <a class="ref" href="graph_2.html#T26" title="GRAPH_2:th.26">GRAPH_2:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_2"><b>suppose </b></a><a NAME="E1:31_2_2"/><i><font color="Green" title="E26">A172</font></i>: 
( <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font> )
 ;<br/><div class="add"><a NAME="E2:31_2_2"/><b>then </b><i><font color="Green" title="E27">A173</font></i>: 
<font color="Maroon" title="c5">m</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> <font color="Maroon" title="c2">vs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:31_2_2"/><b>then </b><i><font color="Green" title="E28">A174</font></i>: 
<font color="Maroon" title="c5">m</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="c3">c</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:31_2_2"/><i><font color="Green" title="E29">A175</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:31_2_2"/><i><font color="Green" title="E30">A176</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c5">m</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="c3">c</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</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="c3">c</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="graph_4.html#E2:31_2_2"><i><font color="Green" title="E27">A173</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">c2</font> = <font color="Maroon" title="c5">m</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">c</font> as  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2"><i><font color="Green" title="E28">A174</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2"><i><font color="Green" title="E29">A175</font></i></a>, <a class="ref" href="graph_4.html#T13" target="_self" title="GRAPH_4:th.13">Th13</a></i>;<br/><b>set </b><font color="Maroon" title="c8">p2</font> = <font color="Maroon" title="c5">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font>;<br/><a NAME="E7:31_2_2"/><i><font color="Green" title="E31">A177</font></i>: 
<font color="Maroon" title="c5">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Maroon" title="c7">c2</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_2"><i><font color="Green" title="E28">A174</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2"><i><font color="Green" title="E29">A175</font></i></a>, <a class="ref" href="graph_4.html#T14" target="_self" title="GRAPH_4:th.14">Th14</a></i>;<br/><b>set </b><font color="Maroon" title="c9">domfc</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c3">c</font> ) </span>}</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/><b>consider </b><font color="Maroon" title="c10">fc</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:31_2_2"/><i><font color="Green" title="E32">A178</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">fc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c3">c</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E10:31_2_2"/><i><font color="Green" title="E33">A179</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</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="c3">c</font> ) </span>}</span>  holds <br/><font color="Maroon" title="c10">fc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E11:31_2_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c3">c</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_2_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c3">c</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">kk</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> <b> such that </b><br/><a NAME="E3:31_2_2_1"/><i><font color="Green" title="E34">A180</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">kk</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">kk</font> &amp; <font color="Maroon" title="c12">kk</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="c3">c</font> )
 ;<br/>
<a NAME="E4:31_2_2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">kk</font> &amp; <font color="Maroon" title="c12">kk</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="c3">c</font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2"><i><font color="Green" title="E29">A175</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_2_1"><i><font color="Green" title="E34">A180</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2_2_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2_1"><i><font color="Green" title="E34">A180</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c11">fc</font> = <font color="Maroon" title="c10">fc</font> as   <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a> <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_2"><i><font color="Green" title="E32">A178</font></i></a>, <a class="ref" href="finseq_1.html#D12" title="FINSEQ_1:def.12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E13:31_2_2"/><i><font color="Green" title="E34">A181</font></i>: 
<font color="Maroon" title="c11">fc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_2_2"/><i><font color="Green" title="E35">A182</font></i>: 
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">fc</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_2_2_2"/><i><font color="Green" title="E36">A183</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:31_2_2_2"/><i><font color="Green" title="E37">A184</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">fc</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">fc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_2_2"><i><font color="Green" title="E35">A182</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_2_2"><i><font color="Green" title="E36">A183</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">kk</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> <b> such that </b><br/><a NAME="E6:31_2_2_2"/><i><font color="Green" title="E38">A185</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">kk</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">kk</font> &amp; <font color="Maroon" title="c15">kk</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="c3">c</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_2"><i><font color="Green" title="E32">A178</font></i></a></i>;<br/>
<a NAME="E7:31_2_2_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">kk</font> &amp; <font color="Maroon" title="c15">kk</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="c3">c</font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2"><i><font color="Green" title="E29">A175</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_2"><i><font color="Green" title="E38">A185</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:31_2_2_2"/><b>then </b><i><font color="Green" title="E39">A186</font></i>: 
<font color="Maroon" title="c13">x</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="c3">c</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_2_2"><i><font color="Green" title="E38">A185</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:31_2_2_2"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">kk</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_2"><i><font color="Green" title="E32">A178</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_2"><i><font color="Green" title="E33">A179</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2_2"><i><font color="Green" title="E37">A184</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_2"><i><font color="Green" title="E38">A185</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:31_2_2_2"/>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2_2"><i><font color="Green" title="E36">A183</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_2"><i><font color="Green" title="E38">A185</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_2_2"><i><font color="Green" title="E39">A186</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c12">fc</font> = <font color="Maroon" title="c11">fc</font> as    <a href="graph_2.html#M1" title="GRAPH_2:mode.1">FinSubsequence</a> of <font color="Maroon" title="c3">c</font> <b>by </b><i><a class="ref" href="graph_2.html#D5" title="GRAPH_2:def.5">GRAPH_2:def 5</a></i>;<br/><b>take </b>
<font color="Maroon" title="c12">fc</font>
;<br/><b>set </b><font color="Maroon" title="c13">domfvs</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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">vs</font> ) </span>}</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/><b>consider </b><font color="Maroon" title="c14">fvs</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E16:31_2_2"/><i><font color="Green" title="E35">A187</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">fvs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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">vs</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E17:31_2_2"/><i><font color="Green" title="E36">A188</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</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">vs</font> ) </span>}</span>  holds <br/><font color="Maroon" title="c14">fvs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E18:31_2_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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">vs</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_2_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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">vs</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">kk</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> <b> such that </b><br/><a NAME="E3:31_2_2_3"/><i><font color="Green" title="E37">A189</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">kk</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">kk</font> &amp; <font color="Maroon" title="c16">kk</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">vs</font> )
 ;<br/>
<a NAME="E4:31_2_2_3"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">kk</font> &amp; <font color="Maroon" title="c16">kk</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">vs</font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2"><i><font color="Green" title="E29">A175</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_2_3"><i><font color="Green" title="E37">A189</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2_2_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2_3"><i><font color="Green" title="E37">A189</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c15">fvs</font> = <font color="Maroon" title="c14">fvs</font> as   <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a> <b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_2"><i><font color="Green" title="E35">A187</font></i></a>, <a class="ref" href="finseq_1.html#D12" title="FINSEQ_1:def.12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E20:31_2_2"/><i><font color="Green" title="E37">A190</font></i>: 
<font color="Maroon" title="c15">fvs</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">vs</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_2_4"/><i><font color="Green" title="E38">A191</font></i>: 
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">fvs</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">x</font>, <font color="Maroon" title="c18">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_2_2_4"/><i><font color="Green" title="E39">A192</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:31_2_2_4"/><i><font color="Green" title="E40">A193</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">fvs</font> &amp; <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">fvs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_2_4"><i><font color="Green" title="E38">A191</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_2_4"><i><font color="Green" title="E39">A192</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">kk</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> <b> such that </b><br/><a NAME="E6:31_2_2_4"/><i><font color="Green" title="E41">A194</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">kk</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">kk</font> &amp; <font color="Maroon" title="c19">kk</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">vs</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_2"><i><font color="Green" title="E35">A187</font></i></a></i>;<br/>
<a NAME="E7:31_2_2_4"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">kk</font> &amp; <font color="Maroon" title="c19">kk</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">vs</font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2"><i><font color="Green" title="E29">A175</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_4"><i><font color="Green" title="E41">A194</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:31_2_2_4"/><b>then </b><i><font color="Green" title="E42">A195</font></i>: 
<font color="Maroon" title="c17">x</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">vs</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_2_4"><i><font color="Green" title="E41">A194</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:31_2_2_4"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">kk</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_2"><i><font color="Green" title="E35">A187</font></i></a>, <a class="txt" href="graph_4.html#E17:31_2_2"><i><font color="Green" title="E36">A188</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2_4"><i><font color="Green" title="E40">A193</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_4"><i><font color="Green" title="E41">A194</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:31_2_2_4"/>
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">vs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2_4"><i><font color="Green" title="E39">A192</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_4"><i><font color="Green" title="E41">A194</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_2_4"><i><font color="Green" title="E42">A195</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c16">fvs</font> = <font color="Maroon" title="c15">fvs</font> as    <a href="graph_2.html#M1" title="GRAPH_2:mode.1">FinSubsequence</a> of <font color="Maroon" title="c2">vs</font> <b>by </b><i><a class="ref" href="graph_2.html#D5" title="GRAPH_2:def.5">GRAPH_2:def 5</a></i>;<br/><b>take </b>
<font color="Maroon" title="c16">fvs</font>
;<br/><b>take </b><font color="Maroon" title="c17">c1</font> = <font color="Maroon" title="c7">c2</font>;<br/><b>take </b><font color="Maroon" title="c18">p1</font> = <font color="Maroon" title="c5">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font>;<br/><a NAME="E22:31_2_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="graph_4.html#E1:31_2_2"><i><font color="Green" title="E26">A172</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green" title="E17">Lm2</font></i></a></i>;<br/><a NAME="E23:31_2_2"/><b>then </b><i><font color="Green" title="E38">A196</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ;<br/><a NAME="E24:31_2_2"/><i><font color="Green" title="E39">A197</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2"><i><font color="Green" title="E28">A174</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E25:31_2_2"/>
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="c5">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2"><i><font color="Green" title="E29">A175</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E26:31_2_2"/><b>then </b>
<font color="Maroon" title="c5">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c19">m1</font> = <font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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>  ;<br/><a NAME="E28:31_2_2"/><i><font color="Green" title="E40">A198</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E29:31_2_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c2</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E23:31_2_2"><i><font color="Green" title="E38">A196</font></i></a>, <a class="txt" href="graph_4.html#E24:31_2_2"><i><font color="Green" title="E39">A197</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c20">lc21</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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>  ;<br/><a NAME="E31:31_2_2"/><i><font color="Green" title="E41">A199</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">lc21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c2</font></span>)</span>
 ;<br/><a NAME="E32:31_2_2"/><i><font color="Green" title="E42">A200</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c5">m</font> <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <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> <font color="Maroon" title="c2">vs</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="graph_4.html#E1:31_2_2"><i><font color="Green" title="E26">A172</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E33:31_2_2"/><b>then </b>
<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="c5">m</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E30"><i><font color="Green" title="E17">Lm2</font></i></a></i>;<br/><a NAME="E34:31_2_2"/><b>then </b><i><font color="Green" title="E43">A201</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ;<br/><a NAME="E35:31_2_2"/><b>then </b><i><font color="Green" title="E44">A202</font></i>: 
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="c5">m</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E23:31_2_2"><i><font color="Green" title="E38">A196</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E36:31_2_2"/><i><font color="Green" title="E45">A203</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E23:31_2_2"><i><font color="Green" title="E38">A196</font></i></a></i>;<br/><a NAME="E37:31_2_2"/>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2"><i><font color="Green" title="E29">A175</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E38:31_2_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E39:31_2_2"/><b>then </b>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60" title="XREAL_1:th.60">XREAL_1:60</a></i>;<br/><b>hence </b><a NAME="E40:31_2_2"/><i><font color="Green" title="E46">A204</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">c1</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> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E36:31_2_2"><i><font color="Green" title="E45">A203</font></i></a>, <a class="ref" href="real_2.html#T173" title="REAL_2:th.173">REAL_2:173</a></i>;<br/><b>thus </b><a NAME="E41:31_2_2"/>
<font color="Maroon" title="c18">p1</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Maroon" title="c17">c1</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="graph_4.html#E5:31_2_2"><i><font color="Green" title="E30">A176</font></i></a>, <a class="ref" href="graph_4.html#T14" target="_self" title="GRAPH_4:th.14">Th14</a></i>;<br/><a NAME="E42:31_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E7:31_2_2"><i><font color="Green" title="E31">A177</font></i></a>, <a class="ref" href="graph_4.html#D5" target="_self" title="GRAPH_4:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E43:31_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">p1</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> <font color="Maroon" title="c2">vs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="graph_4.html#E40:31_2_2"><i><font color="Green" title="E46">A204</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>thus </b><a NAME="E44:31_2_2"/>
<font color="Maroon" title="c2">vs</font> <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="c18">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="graph_4.html#E1:31_2_2"><i><font color="Green" title="E26">A172</font></i></a>, <a class="ref" href="graph_2.html#T12" title="GRAPH_2:th.12">GRAPH_2:12</a></i>;<br/><b>thus </b><a NAME="E45:31_2_2"/>
<font color="Maroon" title="c2">vs</font> <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> <font color="Maroon" title="c2">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">p1</font> <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> <font color="Maroon" title="c18">p1</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="graph_4.html#E1:31_2_2"><i><font color="Green" title="E26">A172</font></i></a>, <a class="ref" href="graph_2.html#T12" title="GRAPH_2:th.12">GRAPH_2:12</a></i>;<br/><a NAME="E46:31_2_2"/><i><font color="Green" title="E47">A205</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c3">c</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_2_5"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c3">c</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c22">k</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> <b> such that </b><br/><a NAME="E3:31_2_2_5"/><i><font color="Green" title="E48">A206</font></i>: 
( <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">k</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">k</font> &amp; <font color="Maroon" title="c22">k</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="c3">c</font> )
 ;<br/>
<a NAME="E4:31_2_2_5"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">k</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2"><i><font color="Green" title="E29">A175</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_2_5"><i><font color="Green" title="E48">A206</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2_2_5"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2_5"><i><font color="Green" title="E48">A206</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c21">domfc</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c3">c</font> ) </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><a NAME="E48:31_2_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c2</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E23:31_2_2"><i><font color="Green" title="E38">A196</font></i></a>, <a class="txt" href="graph_4.html#E24:31_2_2"><i><font color="Green" title="E39">A197</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c22">lc21</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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>  ;<br/><a NAME="E50:31_2_2"/>
<font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c22">lc21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E23:31_2_2"><i><font color="Green" title="E38">A196</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E48">A207</font></i>:  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c21">domfc</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<b>by </b><i><a class="ref" href="graph_2.html#T4" title="GRAPH_2:th.4">GRAPH_2:4</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c2</font>

;<br/><a NAME="E52:31_2_2"/><i><font color="Green" title="E49">A208</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfc</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c21">domfc</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_2"><i><font color="Green" title="E47">A205</font></i></a>, <a class="ref" href="finseq_3.html#T44" title="FINSEQ_3:th.44">FINSEQ_3:44</a></i>;<br/><a NAME="E53:31_2_2"/><i><font color="Green" title="E50">A209</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfc</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">fc</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_2"><i><font color="Green" title="E32">A178</font></i></a>, <a class="txt" href="graph_4.html#E46:31_2_2"><i><font color="Green" title="E47">A205</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/><b>set </b><font color="Maroon" title="c23">SR</font> =  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfc</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c24">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_2_7_1"/><i><font color="Green" title="E51">A210</font></i>: 
<font color="Maroon" title="c24">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">c2</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c25">x</font>, <font color="Maroon" title="c26">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_2_2_7_1"/><i><font color="Green" title="E52">A211</font></i>: 
<font color="Maroon" title="c24">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">x</font>,<font color="Maroon" title="c26">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:31_2_2_7_1"/><i><font color="Green" title="E53">A212</font></i>: 
( <font color="Maroon" title="c25">x</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="c7">c2</font> &amp; <font color="Maroon" title="c26">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_2_7_1"><i><font color="Green" title="E51">A210</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_2_7_1"><i><font color="Green" title="E52">A211</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c27">k</font> = <font color="Maroon" title="c25">x</font> as    <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>  ;<br/><a NAME="E6:31_2_2_7_1"/><i><font color="Green" title="E54">A213</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">k</font> &amp; <font color="Maroon" title="c27">k</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="c7">c2</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2_7_1"><i><font color="Green" title="E53">A212</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:31_2_2_7_1"/><i><font color="Green" title="E55">A214</font></i>: 
<font color="Maroon" title="c25">x</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfc</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E51:31_2_2"><i><font color="Green" title="E48">A207</font></i></a>, <a class="txt" href="graph_4.html#E52:31_2_2"><i><font color="Green" title="E49">A208</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2_7_1"><i><font color="Green" title="E53">A212</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E8:31_2_2_7_1"/><i><font color="Green" title="E56">A215</font></i>: 
<font color="Maroon" title="c19">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c27">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfc</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E23:31_2_2"><i><font color="Green" title="E38">A196</font></i></a>, <a class="txt" href="graph_4.html#E28:31_2_2"><i><font color="Green" title="E40">A198</font></i></a>, <a class="txt" href="graph_4.html#E31:31_2_2"><i><font color="Green" title="E41">A199</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_7_1"><i><font color="Green" title="E54">A213</font></i></a>, <a class="ref" href="graph_2.html#T5" title="GRAPH_2:th.5">GRAPH_2:5</a></i>;<br/><a NAME="E9:31_2_2_7_1"/><i><font color="Green" title="E57">A216</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfc</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfc</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E7:31_2_2_7_1"><i><font color="Green" title="E55">A214</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:31_2_2_7_1"/><b>then </b><i><font color="Green" title="E58">A217</font></i>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfc</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E53:31_2_2"><i><font color="Green" title="E50">A209</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_2_7_1"><i><font color="Green" title="E55">A214</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E11:31_2_2_7_1"/>
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> <font color="Maroon" title="c27">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2_7_1"><i><font color="Green" title="E53">A212</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c28">i</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> <b> such that </b><br/><a NAME="E13:31_2_2_7_1"/><i><font color="Green" title="E59">A218</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">i</font> &amp; <font color="Maroon" title="c28">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> <font color="Maroon" title="c7">c2</font> &amp; <font color="Maroon" title="c27">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_2_7_1"><i><font color="Green" title="E54">A213</font></i></a>, <a class="ref" href="graph_2.html#T1" title="GRAPH_2:th.1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c12">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfc</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> = 
<font color="Maroon" title="c12">fc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c27">k</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31_2_2_7_1"><i><font color="Green" title="E56">A215</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_2_7_1"><i><font color="Green" title="E58">A217</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c28">i</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_2"><i><font color="Green" title="E34">A181</font></i></a>, <a class="txt" href="graph_4.html#E53:31_2_2"><i><font color="Green" title="E50">A209</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_2_7_1"><i><font color="Green" title="E56">A215</font></i></a>, <a class="txt" href="graph_4.html#E9:31_2_2_7_1"><i><font color="Green" title="E57">A216</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a>, <a class="txt" href="graph_4.html#E13:31_2_2_7_1"><i><font color="Green" title="E59">A218</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c26">y</font>
<b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="graph_4.html#E1:31_2_2"><i><font color="Green" title="E26">A172</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2_7_1"><i><font color="Green" title="E53">A212</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_2_7_1"><i><font color="Green" title="E59">A218</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green" title="E17">Lm2</font></i></a></i>
;<br/><b>hence </b><a NAME="E15:31_2_2_7_1"/>
<font color="Maroon" title="c24">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfc</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2_7_1"><i><font color="Green" title="E52">A211</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_2_7_1"><i><font color="Green" title="E58">A217</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_2_7"/><i><font color="Green" title="E51">A219</font></i>: 
<font color="Maroon" title="c24">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfc</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c25">x</font>, <font color="Maroon" title="c26">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:31_2_2_7"/><i><font color="Green" title="E52">A220</font></i>: 
<font color="Maroon" title="c24">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">x</font>,<font color="Maroon" title="c26">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:31_2_2_7"/><i><font color="Green" title="E53">A221</font></i>: 
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfc</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c26">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfc</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_2_7"><i><font color="Green" title="E51">A219</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2_7"><i><font color="Green" title="E52">A220</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E6:31_2_2_7"/><b>then </b><i><font color="Green" title="E54">A222</font></i>: 
( <font color="Maroon" title="c25">x</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfc</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfc</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">fc</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c27">k</font> = <font color="Maroon" title="c25">x</font> as    <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>  ;<br/><a NAME="E8:31_2_2_7"/><i><font color="Green" title="E55">A223</font></i>: 
<font color="Maroon" title="c27">k</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="c7">c2</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E51:31_2_2"><i><font color="Green" title="E48">A207</font></i></a>, <a class="txt" href="graph_4.html#E52:31_2_2"><i><font color="Green" title="E49">A208</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_7"><i><font color="Green" title="E54">A222</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E9:31_2_2_7"/><i><font color="Green" title="E56">A224</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">k</font> &amp; <font color="Maroon" title="c27">k</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="c7">c2</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E51:31_2_2"><i><font color="Green" title="E48">A207</font></i></a>, <a class="txt" href="graph_4.html#E52:31_2_2"><i><font color="Green" title="E49">A208</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_7"><i><font color="Green" title="E54">A222</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:31_2_2_7"/><b>then </b><i><font color="Green" title="E57">A225</font></i>: 
<font color="Maroon" title="c19">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c27">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfc</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E23:31_2_2"><i><font color="Green" title="E38">A196</font></i></a>, <a class="txt" href="graph_4.html#E28:31_2_2"><i><font color="Green" title="E40">A198</font></i></a>, <a class="txt" href="graph_4.html#E31:31_2_2"><i><font color="Green" title="E41">A199</font></i></a>, <a class="ref" href="graph_2.html#T5" title="GRAPH_2:th.5">GRAPH_2:5</a></i>;<br/><a NAME="E11:31_2_2_7"/>
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> <font color="Maroon" title="c27">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_2_7"><i><font color="Green" title="E54">A222</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c28">i</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> <b> such that </b><br/><a NAME="E13:31_2_2_7"/><i><font color="Green" title="E58">A226</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">i</font> &amp; <font color="Maroon" title="c28">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> <font color="Maroon" title="c7">c2</font> &amp; <font color="Maroon" title="c27">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_2_7"><i><font color="Green" title="E56">A224</font></i></a>, <a class="ref" href="graph_2.html#T1" title="GRAPH_2:th.1">GRAPH_2:1</a></i>;<br/><font color="Maroon" title="c7">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font> = 
<font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c28">i</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="graph_4.html#E1:31_2_2"><i><font color="Green" title="E26">A172</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_2_7"><i><font color="Green" title="E58">A226</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green" title="E17">Lm2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c12">fc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfc</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_2_7"><i><font color="Green" title="E58">A226</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_2"><i><font color="Green" title="E34">A181</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_7"><i><font color="Green" title="E54">A222</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_2_7"><i><font color="Green" title="E57">A225</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c26">y</font>
<b>by </b><i><a class="txt" href="graph_4.html#E5:31_2_2_7"><i><font color="Green" title="E53">A221</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_7"><i><font color="Green" title="E54">A222</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E15:31_2_2_7"/>
<font color="Maroon" title="c24">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">c2</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2_7"><i><font color="Green" title="E52">A220</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_2_7"><i><font color="Green" title="E55">A223</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E55:31_2_2"/><b>then </b>
<font color="Maroon" title="c7">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfc</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E56:31_2_2"/>
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c12">fc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">c1</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_2"><i><font color="Green" title="E32">A178</font></i></a>, <a class="ref" href="finseq_1.html#D14" title="FINSEQ_1:def.14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E57:31_2_2"/><i><font color="Green" title="E51">A227</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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">vs</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_2_8"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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">vs</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c25">k</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> <b> such that </b><br/><a NAME="E3:31_2_2_8"/><i><font color="Green" title="E52">A228</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">k</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">k</font> &amp; <font color="Maroon" title="c25">k</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">vs</font> )
 ;<br/>
<a NAME="E4:31_2_2_8"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">k</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2"><i><font color="Green" title="E29">A175</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_2_8"><i><font color="Green" title="E52">A228</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2_2_8"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2_8"><i><font color="Green" title="E52">A228</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c24">domfvs</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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">vs</font> ) </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><a NAME="E59:31_2_2"/>
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> <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="c5">m</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E35:31_2_2"><i><font color="Green" title="E44">A202</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E60:31_2_2"/><b>then </b>
<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="c5">m</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.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="c5">m</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c25">lp21</font> = <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="c5">m</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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>  ;<br/><a NAME="E62:31_2_2"/><i><font color="Green" title="E52">A229</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c25">lp21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E34:31_2_2"><i><font color="Green" title="E43">A201</font></i></a></i>;<br/><a NAME="E63:31_2_2"/><i><font color="Green" title="E53">A230</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c25">lp21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">m1</font> <a href="nat_1.html#K1" title="NAT_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="c5">m</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span></span>)</span>
 ;<br/><i><font color="Green" title="E54">A231</font></i>:  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c24">domfvs</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">m</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="graph_4.html#E62:31_2_2"><i><font color="Green" title="E52">A229</font></i></a>, <a class="ref" href="graph_2.html#T4" title="GRAPH_2:th.4">GRAPH_2:4</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span>

;<br/><a NAME="E65:31_2_2"/><i><font color="Green" title="E55">A232</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c24">domfvs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c24">domfvs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E57:31_2_2"><i><font color="Green" title="E51">A227</font></i></a>, <a class="ref" href="finseq_3.html#T44" title="FINSEQ_3:th.44">FINSEQ_3:44</a></i>;<br/><a NAME="E66:31_2_2"/><i><font color="Green" title="E56">A233</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c24">domfvs</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">fvs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_2"><i><font color="Green" title="E35">A187</font></i></a>, <a class="txt" href="graph_4.html#E57:31_2_2"><i><font color="Green" title="E51">A227</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/><b>set </b><font color="Maroon" title="c26">SR</font> =  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c24">domfvs</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c27">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_2_10_1"/><i><font color="Green" title="E57">A234</font></i>: 
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c28">x</font>, <font color="Maroon" title="c29">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_2_2_10_1"/><i><font color="Green" title="E58">A235</font></i>: 
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c28">x</font>,<font color="Maroon" title="c29">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:31_2_2_10_1"/><i><font color="Green" title="E59">A236</font></i>: 
( <font color="Maroon" title="c28">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span> &amp; <font color="Maroon" title="c29">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_2_10_1"><i><font color="Green" title="E57">A234</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_2_10_1"><i><font color="Green" title="E58">A235</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c30">k</font> = <font color="Maroon" title="c28">x</font> as    <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>  ;<br/><a NAME="E6:31_2_2_10_1"/><i><font color="Green" title="E60">A237</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">k</font> &amp; <font color="Maroon" title="c30">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2_10_1"><i><font color="Green" title="E59">A236</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:31_2_2_10_1"/><i><font color="Green" title="E61">A238</font></i>: 
<font color="Maroon" title="c28">x</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c24">domfvs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E64:31_2_2"><i><font color="Green" title="E54">A231</font></i></a>, <a class="txt" href="graph_4.html#E65:31_2_2"><i><font color="Green" title="E55">A232</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2_10_1"><i><font color="Green" title="E59">A236</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E8:31_2_2_10_1"/><i><font color="Green" title="E62">A239</font></i>: 
<font color="Maroon" title="c19">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c30">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c24">domfvs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="graph_4.html#E28:31_2_2"><i><font color="Green" title="E40">A198</font></i></a>, <a class="txt" href="graph_4.html#E34:31_2_2"><i><font color="Green" title="E43">A201</font></i></a>, <a class="txt" href="graph_4.html#E63:31_2_2"><i><font color="Green" title="E53">A230</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_10_1"><i><font color="Green" title="E60">A237</font></i></a>, <a class="ref" href="graph_2.html#T5" title="GRAPH_2:th.5">GRAPH_2:5</a></i>;<br/><a NAME="E9:31_2_2_10_1"/><i><font color="Green" title="E63">A240</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c24">domfvs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c24">domfvs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E7:31_2_2_10_1"><i><font color="Green" title="E61">A238</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:31_2_2_10_1"/><b>then </b><i><font color="Green" title="E64">A241</font></i>: 
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c24">domfvs</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E66:31_2_2"><i><font color="Green" title="E56">A233</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_2_10_1"><i><font color="Green" title="E61">A238</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E11:31_2_2_10_1"/>
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> <font color="Maroon" title="c30">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2_10_1"><i><font color="Green" title="E59">A236</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c31">i</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> <b> such that </b><br/><a NAME="E13:31_2_2_10_1"/><i><font color="Green" title="E65">A242</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">i</font> &amp; <font color="Maroon" title="c31">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="c5">m</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span> &amp; <font color="Maroon" title="c30">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_2_10_1"><i><font color="Green" title="E60">A237</font></i></a>, <a class="ref" href="graph_2.html#T1" title="GRAPH_2:th.1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c24">domfvs</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> = 
<font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c30">k</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31_2_2_10_1"><i><font color="Green" title="E62">A239</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_2_10_1"><i><font color="Green" title="E64">A241</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">i</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_2"><i><font color="Green" title="E37">A190</font></i></a>, <a class="txt" href="graph_4.html#E66:31_2_2"><i><font color="Green" title="E56">A233</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_2_10_1"><i><font color="Green" title="E62">A239</font></i></a>, <a class="txt" href="graph_4.html#E9:31_2_2_10_1"><i><font color="Green" title="E63">A240</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a>, <a class="txt" href="graph_4.html#E13:31_2_2_10_1"><i><font color="Green" title="E65">A242</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c29">y</font>
<b>by </b><i><a class="txt" href="graph_4.html#E32:31_2_2"><i><font color="Green" title="E42">A200</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2_10_1"><i><font color="Green" title="E59">A236</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_2_10_1"><i><font color="Green" title="E65">A242</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green" title="E17">Lm2</font></i></a></i>
;<br/><b>hence </b><a NAME="E15:31_2_2_10_1"/>
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c24">domfvs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2_10_1"><i><font color="Green" title="E58">A235</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_2_10_1"><i><font color="Green" title="E64">A241</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_2_10"/><i><font color="Green" title="E57">A243</font></i>: 
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c24">domfvs</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c28">x</font>, <font color="Maroon" title="c29">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:31_2_2_10"/><i><font color="Green" title="E58">A244</font></i>: 
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c28">x</font>,<font color="Maroon" title="c29">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:31_2_2_10"/><i><font color="Green" title="E59">A245</font></i>: 
( <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c24">domfvs</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c29">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c24">domfvs</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_2_10"><i><font color="Green" title="E57">A243</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2_10"><i><font color="Green" title="E58">A244</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E6:31_2_2_10"/><b>then </b><i><font color="Green" title="E60">A246</font></i>: 
( <font color="Maroon" title="c28">x</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c24">domfvs</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c24">domfvs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">fvs</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c30">k</font> = <font color="Maroon" title="c28">x</font> as    <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>  ;<br/><a NAME="E8:31_2_2_10"/><i><font color="Green" title="E61">A247</font></i>: 
<font color="Maroon" title="c30">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E64:31_2_2"><i><font color="Green" title="E54">A231</font></i></a>, <a class="txt" href="graph_4.html#E65:31_2_2"><i><font color="Green" title="E55">A232</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_10"><i><font color="Green" title="E60">A246</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E9:31_2_2_10"/><i><font color="Green" title="E62">A248</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">k</font> &amp; <font color="Maroon" title="c30">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_4.html#E64:31_2_2"><i><font color="Green" title="E54">A231</font></i></a>, <a class="txt" href="graph_4.html#E65:31_2_2"><i><font color="Green" title="E55">A232</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_10"><i><font color="Green" title="E60">A246</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:31_2_2_10"/><b>then </b><i><font color="Green" title="E63">A249</font></i>: 
<font color="Maroon" title="c19">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c30">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c24">domfvs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="graph_4.html#E28:31_2_2"><i><font color="Green" title="E40">A198</font></i></a>, <a class="txt" href="graph_4.html#E34:31_2_2"><i><font color="Green" title="E43">A201</font></i></a>, <a class="txt" href="graph_4.html#E63:31_2_2"><i><font color="Green" title="E53">A230</font></i></a>, <a class="ref" href="graph_2.html#T5" title="GRAPH_2:th.5">GRAPH_2:5</a></i>;<br/><a NAME="E11:31_2_2_10"/>
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> <font color="Maroon" title="c30">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_2_10"><i><font color="Green" title="E60">A246</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c31">i</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> <b> such that </b><br/><a NAME="E13:31_2_2_10"/><i><font color="Green" title="E64">A250</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">i</font> &amp; <font color="Maroon" title="c31">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="c5">m</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span> &amp; <font color="Maroon" title="c30">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_2_10"><i><font color="Green" title="E62">A248</font></i></a>, <a class="ref" href="graph_2.html#T1" title="GRAPH_2:th.1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">k</font> = 
<font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">i</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E32:31_2_2"><i><font color="Green" title="E42">A200</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_2_10"><i><font color="Green" title="E64">A250</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green" title="E17">Lm2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c24">domfvs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">k</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_2_10"><i><font color="Green" title="E64">A250</font></i></a>, <a class="txt" href="graph_4.html#E20:31_2_2"><i><font color="Green" title="E37">A190</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_10"><i><font color="Green" title="E60">A246</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_2_10"><i><font color="Green" title="E63">A249</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c29">y</font>
<b>by </b><i><a class="txt" href="graph_4.html#E5:31_2_2_10"><i><font color="Green" title="E59">A245</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_10"><i><font color="Green" title="E60">A246</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E15:31_2_2_10"/>
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2_10"><i><font color="Green" title="E58">A244</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_2_10"><i><font color="Green" title="E61">A247</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E68:31_2_2"/><b>then </b>
<font color="Maroon" title="c5">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c24">domfvs</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E69:31_2_2"/>
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c16">fvs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">p1</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_2"><i><font color="Green" title="E35">A187</font></i></a>, <a class="ref" href="finseq_1.html#D14" title="FINSEQ_1:def.14">FINSEQ_1:def 14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_3"><b>suppose </b></a><a NAME="E1:31_2_3"/><i><font color="Green" title="E26">A251</font></i>: 
( <font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font> )
 ;<br/><div class="add"><a NAME="E2:31_2_3"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:31_2_3"/><b>then </b>
1 <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="c4">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:31_2_3"/><b>then </b><i><font color="Green" title="E27">A252</font></i>: 
<span class="p1">(<span class="default">1 <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> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E5:31_2_3"/>
<font color="Maroon" title="c4">n</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> <font color="Maroon" title="c2">vs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:31_2_3"/><b>then </b><i><font color="Green" title="E28">A253</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span> <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> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green" title="E21">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="E7:31_2_3"/><i><font color="Green" title="E29">A254</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">c1</font> = 1,<font color="Maroon" title="c6">n1</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">c</font> as  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_3"><i><font color="Green" title="E27">A252</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green" title="E28">A253</font></i></a>, <a class="ref" href="graph_4.html#T13" target="_self" title="GRAPH_4:th.13">Th13</a></i>;<br/><b>set </b><font color="Maroon" title="c8">pp</font> = 1,<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">vs</font>;<br/><a NAME="E9:31_2_3"/><i><font color="Green" title="E30">A255</font></i>: 
( 1 <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">vs</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:31_2_3"/><i><font color="Green" title="E31">A256</font></i>: 
1,<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">vs</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Maroon" title="c7">c1</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="graph_4.html#E9:31"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_3"><i><font color="Green" title="E27">A252</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green" title="E28">A253</font></i></a>, <a class="ref" href="graph_4.html#T14" target="_self" title="GRAPH_4:th.14">Th14</a></i>;<br/><a NAME="E11:31_2_3"/><b>then </b><i><font color="Green" title="E32">A257</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default">1,<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">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="graph_4.html#D5" target="_self" title="GRAPH_4:def.5">Def5</a></i>;<br/><b>set </b><font color="Maroon" title="c9">domfc</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/><b>consider </b><font color="Maroon" title="c10">fc</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E13:31_2_3"/><i><font color="Green" title="E33">A258</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">fc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E14:31_2_3"/><i><font color="Green" title="E34">A259</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span>  holds <br/><font color="Maroon" title="c10">fc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E15:31_2_3"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_3_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">kk</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> <b> such that </b><br/><a NAME="E3:31_2_3_1"/><i><font color="Green" title="E35">A260</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">kk</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">kk</font> &amp; <font color="Maroon" title="c12">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> )
 ;<br/>
<a NAME="E4:31_2_3_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">kk</font> &amp; <font color="Maroon" title="c12">kk</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="c3">c</font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green" title="E28">A253</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_3_1"><i><font color="Green" title="E35">A260</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2_3_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_3_1"><i><font color="Green" title="E35">A260</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c11">fc</font> = <font color="Maroon" title="c10">fc</font> as   <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a> <b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_3"><i><font color="Green" title="E33">A258</font></i></a>, <a class="ref" href="finseq_1.html#D12" title="FINSEQ_1:def.12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E17:31_2_3"/><i><font color="Green" title="E35">A261</font></i>: 
<font color="Maroon" title="c11">fc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_3_2"/><i><font color="Green" title="E36">A262</font></i>: 
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">fc</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_2_3_2"/><i><font color="Green" title="E37">A263</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:31_2_3_2"/><i><font color="Green" title="E38">A264</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">fc</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">fc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_3_2"><i><font color="Green" title="E36">A262</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_3_2"><i><font color="Green" title="E37">A263</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">kk</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> <b> such that </b><br/><a NAME="E6:31_2_3_2"/><i><font color="Green" title="E39">A265</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">kk</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">kk</font> &amp; <font color="Maroon" title="c15">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_3"><i><font color="Green" title="E33">A258</font></i></a></i>;<br/>
<a NAME="E7:31_2_3_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">kk</font> &amp; <font color="Maroon" title="c15">kk</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="c3">c</font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green" title="E28">A253</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_2"><i><font color="Green" title="E39">A265</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:31_2_3_2"/><b>then </b><i><font color="Green" title="E40">A266</font></i>: 
<font color="Maroon" title="c13">x</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="c3">c</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_3_2"><i><font color="Green" title="E39">A265</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:31_2_3_2"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">kk</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_3"><i><font color="Green" title="E33">A258</font></i></a>, <a class="txt" href="graph_4.html#E14:31_2_3"><i><font color="Green" title="E34">A259</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_3_2"><i><font color="Green" title="E38">A264</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_2"><i><font color="Green" title="E39">A265</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:31_2_3_2"/>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_3_2"><i><font color="Green" title="E37">A263</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_2"><i><font color="Green" title="E39">A265</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_3_2"><i><font color="Green" title="E40">A266</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c12">fc</font> = <font color="Maroon" title="c11">fc</font> as    <a href="graph_2.html#M1" title="GRAPH_2:mode.1">FinSubsequence</a> of <font color="Maroon" title="c3">c</font> <b>by </b><i><a class="ref" href="graph_2.html#D5" title="GRAPH_2:def.5">GRAPH_2:def 5</a></i>;<br/><b>take </b>
<font color="Maroon" title="c12">fc</font>
;<br/><b>set </b><font color="Maroon" title="c13">domfvs</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/><b>consider </b><font color="Maroon" title="c14">fvs</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E20:31_2_3"/><i><font color="Green" title="E36">A267</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">fvs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E21:31_2_3"/><i><font color="Green" title="E37">A268</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span>  holds <br/><font color="Maroon" title="c14">fvs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E22:31_2_3"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_3_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">kk</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> <b> such that </b><br/><a NAME="E3:31_2_3_3"/><i><font color="Green" title="E38">A269</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">kk</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">kk</font> &amp; <font color="Maroon" title="c16">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> )
 ;<br/>
<a NAME="E4:31_2_3_3"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">kk</font> &amp; <font color="Maroon" title="c16">kk</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">vs</font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_3"><i><font color="Green" title="E30">A255</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_3_3"><i><font color="Green" title="E38">A269</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2_3_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_3_3"><i><font color="Green" title="E38">A269</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c15">fvs</font> = <font color="Maroon" title="c14">fvs</font> as   <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a> <b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_3"><i><font color="Green" title="E36">A267</font></i></a>, <a class="ref" href="finseq_1.html#D12" title="FINSEQ_1:def.12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E24:31_2_3"/><i><font color="Green" title="E38">A270</font></i>: 
<font color="Maroon" title="c15">fvs</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">vs</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_3_4"/><i><font color="Green" title="E39">A271</font></i>: 
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">fvs</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">x</font>, <font color="Maroon" title="c18">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_2_3_4"/><i><font color="Green" title="E40">A272</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:31_2_3_4"/><i><font color="Green" title="E41">A273</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">fvs</font> &amp; <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">fvs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_3_4"><i><font color="Green" title="E39">A271</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_3_4"><i><font color="Green" title="E40">A272</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">kk</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> <b> such that </b><br/><a NAME="E6:31_2_3_4"/><i><font color="Green" title="E42">A274</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">kk</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">kk</font> &amp; <font color="Maroon" title="c19">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_3"><i><font color="Green" title="E36">A267</font></i></a></i>;<br/>
<a NAME="E7:31_2_3_4"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">kk</font> &amp; <font color="Maroon" title="c19">kk</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">vs</font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_3"><i><font color="Green" title="E30">A255</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_4"><i><font color="Green" title="E42">A274</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:31_2_3_4"/><b>then </b><i><font color="Green" title="E43">A275</font></i>: 
<font color="Maroon" title="c17">x</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">vs</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_3_4"><i><font color="Green" title="E42">A274</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:31_2_3_4"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">kk</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_3"><i><font color="Green" title="E36">A267</font></i></a>, <a class="txt" href="graph_4.html#E21:31_2_3"><i><font color="Green" title="E37">A268</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_3_4"><i><font color="Green" title="E41">A273</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_4"><i><font color="Green" title="E42">A274</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:31_2_3_4"/>
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">vs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_3_4"><i><font color="Green" title="E40">A272</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_4"><i><font color="Green" title="E42">A274</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_3_4"><i><font color="Green" title="E43">A275</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c16">fvs</font> = <font color="Maroon" title="c15">fvs</font> as    <a href="graph_2.html#M1" title="GRAPH_2:mode.1">FinSubsequence</a> of <font color="Maroon" title="c2">vs</font> <b>by </b><i><a class="ref" href="graph_2.html#D5" title="GRAPH_2:def.5">GRAPH_2:def 5</a></i>;<br/><b>take </b>
<font color="Maroon" title="c16">fvs</font>
;<br/><b>take </b><font color="Maroon" title="c17">c'</font> = <font color="Maroon" title="c7">c1</font>;<br/><b>take </b><font color="Maroon" title="c18">p1</font> = 1,<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">vs</font>;<br/><a NAME="E26:31_2_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green" title="E28">A253</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_3"><i><font color="Green" title="E29">A254</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green" title="E17">Lm2</font></i></a></i>;<br/><a NAME="E27:31_2_3"/><b>then </b><i><font color="Green" title="E39">A276</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E7:31"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><b>hence </b><a NAME="E28:31_2_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">c'</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> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green" title="E28">A253</font></i></a></i>;<br/><b>thus </b><a NAME="E29:31_2_3"/>
<font color="Maroon" title="c18">p1</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Maroon" title="c17">c'</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="graph_4.html#E9:31"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_3"><i><font color="Green" title="E27">A252</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green" title="E28">A253</font></i></a>, <a class="ref" href="graph_4.html#T14" target="_self" title="GRAPH_4:th.14">Th14</a></i>;<br/><a NAME="E30:31_2_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_3"><i><font color="Green" title="E31">A256</font></i></a>, <a class="ref" href="graph_4.html#D5" target="_self" title="GRAPH_4:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E31:31_2_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">p1</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> <font color="Maroon" title="c2">vs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green" title="E39">A276</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>thus </b><a NAME="E32:31_2_3"/>
<font color="Maroon" title="c2">vs</font> <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="c18">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_3"><i><font color="Green" title="E30">A255</font></i></a>, <a class="ref" href="graph_2.html#T12" title="GRAPH_2:th.12">GRAPH_2:12</a></i>;<br/><b>thus </b><a NAME="E33:31_2_3"/>
<font color="Maroon" title="c2">vs</font> <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> <font color="Maroon" title="c2">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">p1</font> <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> <font color="Maroon" title="c18">p1</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graph_4.html#E1:31_2_3"><i><font color="Green" title="E26">A251</font></i></a>, <a class="ref" href="graph_2.html#T12" title="GRAPH_2:th.12">GRAPH_2:12</a></i>;<br/><a NAME="E34:31_2_3"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_3_5"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c20">k</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> <b> such that </b><br/><a NAME="E3:31_2_3_5"/><i><font color="Green" title="E40">A277</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">k</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">k</font> &amp; <font color="Maroon" title="c20">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> )
 ;<br/>
<a NAME="E4:31_2_3_5"/>
<font color="Maroon" title="c20">k</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="c3">c</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green" title="E28">A253</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_3_5"><i><font color="Green" title="E40">A277</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2_3_5"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_3_5"><i><font color="Green" title="E40">A277</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c19">domfc</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><a NAME="E36:31_2_3"/>
<font color="Maroon" title="c19">domfc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D1" title="FINSEQ_1:def.1">FINSEQ_1:def 1</a></i>;<br/><a NAME="E37:31_2_3"/><b>then </b><i><font color="Green" title="E40">A278</font></i>: 
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c19">domfc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c6">n1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T54" title="FINSEQ_3:th.54">FINSEQ_3:54</a></i>;<br/><a NAME="E38:31_2_3"/><b>then </b><i><font color="Green" title="E41">A279</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c19">domfc</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n1</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/><b>set </b><font color="Maroon" title="c20">SL</font> =  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c19">domfc</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_3_6_1"/><i><font color="Green" title="E42">A280</font></i>: 
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">c1</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c22">x</font>, <font color="Maroon" title="c23">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_2_3_6_1"/><i><font color="Green" title="E43">A281</font></i>: 
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">x</font>,<font color="Maroon" title="c23">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:31_2_3_6_1"/><i><font color="Green" title="E44">A282</font></i>: 
( <font color="Maroon" title="c22">x</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="c7">c1</font> &amp; <font color="Maroon" title="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_3_6_1"><i><font color="Green" title="E42">A280</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_3_6_1"><i><font color="Green" title="E43">A281</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c24">k</font> = <font color="Maroon" title="c22">x</font> as    <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>  ;<br/><a NAME="E6:31_2_3_6_1"/><i><font color="Green" title="E45">A283</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">k</font> &amp; <font color="Maroon" title="c24">k</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="c7">c1</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_3_6_1"><i><font color="Green" title="E44">A282</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:31_2_3_6_1"/><b>then </b><i><font color="Green" title="E46">A284</font></i>: 
<font color="Maroon" title="c22">x</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c19">domfc</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green" title="E39">A276</font></i></a>, <a class="txt" href="graph_4.html#E38:31_2_3"><i><font color="Green" title="E41">A279</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:31_2_3_6_1"/><b>then </b><i><font color="Green" title="E47">A285</font></i>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c19">domfc</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E37:31_2_3"><i><font color="Green" title="E40">A278</font></i></a>, <a class="txt" href="graph_4.html#E38:31_2_3"><i><font color="Green" title="E41">A279</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><a NAME="E9:31_2_3_6_1"/><i><font color="Green" title="E48">A286</font></i>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">domfc</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green" title="E39">A276</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_6_1"><i><font color="Green" title="E45">A283</font></i></a></i>;<br/><a NAME="E10:31_2_3_6_1"/><b>then </b><i><font color="Green" title="E49">A287</font></i>: 
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c19">domfc</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_3"><i><font color="Green" title="E33">A258</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_3_6_1"><i><font color="Green" title="E46">A284</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_3_6_1"><i><font color="Green" title="E47">A285</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E11:31_2_3_6_1"/>
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> <font color="Maroon" title="c24">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_3_6_1"><i><font color="Green" title="E44">A282</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c25">i</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> <b> such that </b><br/><a NAME="E13:31_2_3_6_1"/><i><font color="Green" title="E50">A288</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">i</font> &amp; <font color="Maroon" title="c25">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n1</font> &amp; <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green" title="E39">A276</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_6_1"><i><font color="Green" title="E45">A283</font></i></a>, <a class="ref" href="graph_2.html#T1" title="GRAPH_2:th.1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c12">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c19">domfc</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> = 
<font color="Maroon" title="c12">fc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31_2_3_6_1"><i><font color="Green" title="E47">A285</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_6_1"><i><font color="Green" title="E49">A287</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c25">i</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_3"><i><font color="Green" title="E33">A258</font></i></a>, <a class="txt" href="graph_4.html#E17:31_2_3"><i><font color="Green" title="E35">A261</font></i></a>, <a class="txt" href="graph_4.html#E9:31_2_3_6_1"><i><font color="Green" title="E48">A286</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_3_6_1"><i><font color="Green" title="E50">A288</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c23">y</font>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green" title="E28">A253</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_3"><i><font color="Green" title="E29">A254</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green" title="E39">A276</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_3_6_1"><i><font color="Green" title="E44">A282</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_3_6_1"><i><font color="Green" title="E50">A288</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green" title="E17">Lm2</font></i></a></i>
;<br/><b>hence </b><a NAME="E15:31_2_3_6_1"/>
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c19">domfc</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_3_6_1"><i><font color="Green" title="E43">A281</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_6_1"><i><font color="Green" title="E49">A287</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_3_6"/><i><font color="Green" title="E42">A289</font></i>: 
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c19">domfc</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c22">x</font>, <font color="Maroon" title="c23">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:31_2_3_6"/><i><font color="Green" title="E43">A290</font></i>: 
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">x</font>,<font color="Maroon" title="c23">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:31_2_3_6"/><i><font color="Green" title="E44">A291</font></i>: 
( <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c19">domfc</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c19">domfc</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_3_6"><i><font color="Green" title="E42">A289</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_3_6"><i><font color="Green" title="E43">A290</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E6:31_2_3_6"/><b>then </b><i><font color="Green" title="E45">A292</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c19">domfc</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">fc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c19">domfc</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E7:31_2_3_6"/><i><font color="Green" title="E46">A293</font></i>: 
( <font color="Maroon" title="c22">x</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c19">domfc</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c19">domfc</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">fc</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E5:31_2_3_6"><i><font color="Green" title="E44">A291</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E8:31_2_3_6"/><b>then </b>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E38:31_2_3"><i><font color="Green" title="E41">A279</font></i></a>, <a class="ref" href="finseq_1.html#D1" title="FINSEQ_1:def.1">FINSEQ_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">k</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> <b> such that </b><br/><a NAME="E10:31_2_3_6"/><i><font color="Green" title="E47">A294</font></i>: 
( <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">k</font> &amp; <font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> )
 ;<br/><a NAME="E11:31_2_3_6"/><i><font color="Green" title="E48">A295</font></i>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">fc</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_3"><i><font color="Green" title="E33">A258</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_6"><i><font color="Green" title="E47">A294</font></i></a></i>;<br/><a NAME="E12:31_2_3_6"/><i><font color="Green" title="E49">A296</font></i>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c19">domfc</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E37:31_2_3"><i><font color="Green" title="E40">A278</font></i></a>, <a class="txt" href="graph_4.html#E38:31_2_3"><i><font color="Green" title="E41">A279</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_3_6"><i><font color="Green" title="E46">A293</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_6"><i><font color="Green" title="E47">A294</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><a NAME="E13:31_2_3_6"/><i><font color="Green" title="E50">A297</font></i>: 
<font color="Maroon" title="c24">k</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="c7">c1</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green" title="E39">A276</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_6"><i><font color="Green" title="E47">A294</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:31_2_3_6"/>
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> <font color="Maroon" title="c24">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_3_6"><i><font color="Green" title="E47">A294</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c25">i</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> <b> such that </b><br/><a NAME="E16:31_2_3_6"/><i><font color="Green" title="E51">A298</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">i</font> &amp; <font color="Maroon" title="c25">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n1</font> &amp; <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_3_6"><i><font color="Green" title="E47">A294</font></i></a>, <a class="ref" href="graph_2.html#T1" title="GRAPH_2:th.1">GRAPH_2:1</a></i>;<br/><font color="Maroon" title="c7">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> = 
<font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green" title="E28">A253</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_3"><i><font color="Green" title="E29">A254</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green" title="E39">A276</font></i></a>, <a class="txt" href="graph_4.html#E16:31_2_3_6"><i><font color="Green" title="E51">A298</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green" title="E17">Lm2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c23">y</font>
<b>by </b><i><a class="txt" href="graph_4.html#E17:31_2_3"><i><font color="Green" title="E35">A261</font></i></a>, <a class="txt" href="graph_4.html#E5:31_2_3_6"><i><font color="Green" title="E44">A291</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_6"><i><font color="Green" title="E45">A292</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_6"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="graph_4.html#E11:31_2_3_6"><i><font color="Green" title="E48">A295</font></i></a>, <a class="txt" href="graph_4.html#E12:31_2_3_6"><i><font color="Green" title="E49">A296</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
;<br/><b>hence </b><a NAME="E18:31_2_3_6"/>
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">c1</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_3_6"><i><font color="Green" title="E43">A290</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_6"><i><font color="Green" title="E47">A294</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_3_6"><i><font color="Green" title="E50">A297</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E40:31_2_3"/><b>then </b>
<font color="Maroon" title="c7">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">fc</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c19">domfc</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E41:31_2_3"/>
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c12">fc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">c'</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_3"><i><font color="Green" title="E33">A258</font></i></a>, <a class="ref" href="finseq_1.html#D14" title="FINSEQ_1:def.14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E42:31_2_3"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_3_7"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c22">k</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> <b> such that </b><br/><a NAME="E3:31_2_3_7"/><i><font color="Green" title="E42">A299</font></i>: 
( <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">k</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">k</font> &amp; <font color="Maroon" title="c22">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> )
 ;<br/>
<a NAME="E4:31_2_3_7"/>
<font color="Maroon" title="c22">k</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">vs</font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_3"><i><font color="Green" title="E30">A255</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_3_7"><i><font color="Green" title="E42">A299</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2_3_7"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_3_7"><i><font color="Green" title="E42">A299</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c21">domfvs</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><a NAME="E44:31_2_3"/>
<font color="Maroon" title="c21">domfvs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D1" title="FINSEQ_1:def.1">FINSEQ_1:def 1</a></i>;<br/><a NAME="E45:31_2_3"/><b>then </b><i><font color="Green" title="E42">A300</font></i>: 
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfvs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T54" title="FINSEQ_3:th.54">FINSEQ_3:54</a></i>;<br/><a NAME="E46:31_2_3"/><b>then </b><i><font color="Green" title="E43">A301</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfvs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/><b>set </b><font color="Maroon" title="c22">SL</font> =  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfvs</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_3_8_1"/><i><font color="Green" title="E44">A302</font></i>: 
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1,<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">vs</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c24">x</font>, <font color="Maroon" title="c25">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_2_3_8_1"/><i><font color="Green" title="E45">A303</font></i>: 
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c25">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:31_2_3_8_1"/><i><font color="Green" title="E46">A304</font></i>: 
( <font color="Maroon" title="c24">x</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> <span class="p1">(<span class="default">1,<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">vs</font></span>)</span> &amp; <font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1,<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">vs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_3_8_1"><i><font color="Green" title="E44">A302</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_3_8_1"><i><font color="Green" title="E45">A303</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c26">k</font> = <font color="Maroon" title="c24">x</font> as    <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>  ;<br/><a NAME="E6:31_2_3_8_1"/><i><font color="Green" title="E47">A305</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">k</font> &amp; <font color="Maroon" title="c26">k</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> <span class="p1">(<span class="default">1,<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">vs</font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_3_8_1"><i><font color="Green" title="E46">A304</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:31_2_3_8_1"/><b>then </b><i><font color="Green" title="E48">A306</font></i>: 
<font color="Maroon" title="c24">x</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfvs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E11:31_2_3"><i><font color="Green" title="E32">A257</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green" title="E39">A276</font></i></a>, <a class="txt" href="graph_4.html#E46:31_2_3"><i><font color="Green" title="E43">A301</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:31_2_3_8_1"/><b>then </b><i><font color="Green" title="E49">A307</font></i>: 
<font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfvs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E45:31_2_3"><i><font color="Green" title="E42">A300</font></i></a>, <a class="txt" href="graph_4.html#E46:31_2_3"><i><font color="Green" title="E43">A301</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><a NAME="E9:31_2_3_8_1"/><i><font color="Green" title="E50">A308</font></i>: 
<font color="Maroon" title="c26">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">domfvs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E11:31_2_3"><i><font color="Green" title="E32">A257</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green" title="E39">A276</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_8_1"><i><font color="Green" title="E47">A305</font></i></a></i>;<br/><a NAME="E10:31_2_3_8_1"/><b>then </b><i><font color="Green" title="E51">A309</font></i>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfvs</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_3"><i><font color="Green" title="E36">A267</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_3_8_1"><i><font color="Green" title="E48">A306</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_3_8_1"><i><font color="Green" title="E49">A307</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E11:31_2_3_8_1"/>
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> <font color="Maroon" title="c26">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_3_8_1"><i><font color="Green" title="E46">A304</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c27">i</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> <b> such that </b><br/><a NAME="E13:31_2_3_8_1"/><i><font color="Green" title="E52">A310</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> &amp; <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E11:31_2_3"><i><font color="Green" title="E32">A257</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green" title="E39">A276</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_8_1"><i><font color="Green" title="E47">A305</font></i></a>, <a class="ref" href="graph_2.html#T1" title="GRAPH_2:th.1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfvs</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> = 
<font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31_2_3_8_1"><i><font color="Green" title="E49">A307</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_8_1"><i><font color="Green" title="E51">A309</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c27">i</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_3"><i><font color="Green" title="E36">A267</font></i></a>, <a class="txt" href="graph_4.html#E24:31_2_3"><i><font color="Green" title="E38">A270</font></i></a>, <a class="txt" href="graph_4.html#E9:31_2_3_8_1"><i><font color="Green" title="E50">A308</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_3_8_1"><i><font color="Green" title="E52">A310</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c25">y</font>
<b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_3"><i><font color="Green" title="E30">A255</font></i></a>, <a class="txt" href="graph_4.html#E11:31_2_3"><i><font color="Green" title="E32">A257</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green" title="E39">A276</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_3_8_1"><i><font color="Green" title="E46">A304</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_3_8_1"><i><font color="Green" title="E52">A310</font></i></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></i>
;<br/><b>hence </b><a NAME="E15:31_2_3_8_1"/>
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfvs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_3_8_1"><i><font color="Green" title="E45">A303</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_8_1"><i><font color="Green" title="E51">A309</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_3_8"/><i><font color="Green" title="E44">A311</font></i>: 
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfvs</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c24">x</font>, <font color="Maroon" title="c25">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:31_2_3_8"/><i><font color="Green" title="E45">A312</font></i>: 
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c25">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:31_2_3_8"/><i><font color="Green" title="E46">A313</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfvs</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfvs</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_3_8"><i><font color="Green" title="E44">A311</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_3_8"><i><font color="Green" title="E45">A312</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E6:31_2_3_8"/><b>then </b><i><font color="Green" title="E47">A314</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfvs</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfvs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E7:31_2_3_8"/><i><font color="Green" title="E48">A315</font></i>: 
( <font color="Maroon" title="c24">x</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfvs</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfvs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">fvs</font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E5:31_2_3_8"><i><font color="Green" title="E46">A313</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E8:31_2_3_8"/><b>then </b>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_3"><i><font color="Green" title="E43">A301</font></i></a>, <a class="ref" href="finseq_1.html#D1" title="FINSEQ_1:def.1">FINSEQ_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c26">k</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> <b> such that </b><br/><a NAME="E10:31_2_3_8"/><i><font color="Green" title="E49">A316</font></i>: 
( <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">k</font> &amp; <font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> )
 ;<br/><a NAME="E11:31_2_3_8"/><i><font color="Green" title="E50">A317</font></i>: 
<font color="Maroon" title="c26">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">fvs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_3"><i><font color="Green" title="E36">A267</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_8"><i><font color="Green" title="E49">A316</font></i></a></i>;<br/><a NAME="E12:31_2_3_8"/><i><font color="Green" title="E51">A318</font></i>: 
<font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfvs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E45:31_2_3"><i><font color="Green" title="E42">A300</font></i></a>, <a class="txt" href="graph_4.html#E46:31_2_3"><i><font color="Green" title="E43">A301</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_3_8"><i><font color="Green" title="E48">A315</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_8"><i><font color="Green" title="E49">A316</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><a NAME="E13:31_2_3_8"/><i><font color="Green" title="E52">A319</font></i>: 
<font color="Maroon" title="c26">k</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> <span class="p1">(<span class="default">1,<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">vs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E11:31_2_3"><i><font color="Green" title="E32">A257</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green" title="E39">A276</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_8"><i><font color="Green" title="E49">A316</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:31_2_3_8"/>
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> <font color="Maroon" title="c26">k</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_3_8"><i><font color="Green" title="E49">A316</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c27">i</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> <b> such that </b><br/><a NAME="E16:31_2_3_8"/><i><font color="Green" title="E53">A320</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> &amp; <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_3_8"><i><font color="Green" title="E49">A316</font></i></a>, <a class="ref" href="graph_2.html#T1" title="GRAPH_2:th.1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default">1,<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">vs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font> = 
<font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font>
<b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_3"><i><font color="Green" title="E30">A255</font></i></a>, <a class="txt" href="graph_4.html#E11:31_2_3"><i><font color="Green" title="E32">A257</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green" title="E39">A276</font></i></a>, <a class="txt" href="graph_4.html#E16:31_2_3_8"><i><font color="Green" title="E53">A320</font></i></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c25">y</font>
<b>by </b><i><a class="txt" href="graph_4.html#E24:31_2_3"><i><font color="Green" title="E38">A270</font></i></a>, <a class="txt" href="graph_4.html#E5:31_2_3_8"><i><font color="Green" title="E46">A313</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_8"><i><font color="Green" title="E47">A314</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_8"><i><font color="Green" title="E49">A316</font></i></a>, <a class="txt" href="graph_4.html#E11:31_2_3_8"><i><font color="Green" title="E50">A317</font></i></a>, <a class="txt" href="graph_4.html#E12:31_2_3_8"><i><font color="Green" title="E51">A318</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
;<br/><b>hence </b><a NAME="E18:31_2_3_8"/>
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1,<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">vs</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_3_8"><i><font color="Green" title="E45">A312</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_8"><i><font color="Green" title="E49">A316</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_3_8"><i><font color="Green" title="E52">A319</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E48:31_2_3"/><b>then </b>
1,<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">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">fvs</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c21">domfvs</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E49:31_2_3"/>
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c16">fvs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">p1</font>
 <b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_3"><i><font color="Green" title="E36">A267</font></i></a>, <a class="ref" href="finseq_1.html#D14" title="FINSEQ_1:def.14">FINSEQ_1:def 14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
