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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">vs1</font>, <font color="Olive" title="b2">vs2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font><br/>  for <font color="Olive" title="b3">c1</font>, <font color="Olive" title="b4">c2</font> being    <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">vs1</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Olive" title="b3">c1</font> &amp; <font color="Olive" title="b2">vs2</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Olive" title="b4">c2</font> &amp; <font color="Olive" title="b1">vs1</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="Olive" title="b1">vs1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">vs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 holds <br/><font color="Olive" title="b3">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b4">c2</font> is    <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">vs1</font>, <font color="Maroon" title="c3">vs2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">c1</font>, <font color="Olive" title="b2">c2</font> being    <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font>  st <font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Olive" title="b1">c1</font> &amp; <font color="Maroon" title="c3">vs2</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Olive" title="b2">c2</font> &amp; <font color="Maroon" title="c2">vs1</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">vs1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 holds <br/><font color="Olive" title="b1">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">c2</font> is    <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">c1</font>, <font color="Maroon" title="c5">c2</font> be    <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c4">c1</font> &amp; <font color="Maroon" title="c3">vs2</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c5">c2</font> &amp; <font color="Maroon" title="c2">vs1</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">vs1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 implies <font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font> is    <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:59"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
<font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c4">c1</font>
 <span class="kw">and </span><br/><a NAME="E2:59"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
<font color="Maroon" title="c3">vs2</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c5">c2</font>
 <span class="kw">and </span><br/><a NAME="E3:59"/><span class="lab"><font color="Green" title="E54">A3</font></span>: 
<font color="Maroon" title="c2">vs1</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">vs1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font> is    <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">q</font> = <font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font>;<br/>
<a NAME="E4:59"/><span class="lab"><font color="Green" title="E55">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">vs2</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="c5">c2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="graph_2.html#D6" target="_self" title="GRAPH_2:def.6">Def7</a></span>;<br/>
<a NAME="E5:59"/><span class="kw">then </span>
<font color="Maroon" title="c3">vs2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E6:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A5</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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">vs2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#T13" target="_self" title="GRAPH_2:th.13">Th13</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E57">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:59_1"/><span class="lab"><font color="Green" title="E58">A7</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">and </span><br/><a NAME="E2:59_1"/><span class="lab"><font color="Green" title="E59">A8</font></span>: 
<font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:59_1_1"/>
( <font color="Maroon" title="c8">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">vs1</font> or <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:59_1_1_1"/><span class="lab"><font color="Green" title="E60">A9</font></span>: 
<font color="Maroon" title="c8">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">vs1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span><br/><div class="add"><a NAME="E2:59_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A10</font></span>: 
<font color="Maroon" title="c8">n</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">vs1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E3:59_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="txt" href="#E1:59_1_1_1"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="ref" href="graph_2.html#T14" target="_self" title="GRAPH_2:th.14">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E4:59_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1_1_1"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:59_1_1_2"/><span class="lab"><font color="Green" title="E60">A11</font></span>: 
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:59_1_1_2"/><span class="lab"><font color="Green" title="E61">A12</font></span>: 
<font color="Maroon" title="c8">n</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="c2">vs1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">m</font> = <font color="Maroon" title="c9">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E5:59_1_1_2"/>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1_1_2"><span class="lab"><font color="Green" title="E60">A11</font></span></a>, <a class="txt" href="#E3:59_1_1_2"><span class="lab"><font color="Green" title="E61">A12</font></span></a></span>;<br/><a NAME="E6:59_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A13</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:59_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">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="c2">vs1</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E3:59_1_1_2"><span class="lab"><font color="Green" title="E61">A12</font></span></a></span>;<br/><a NAME="E8:59_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">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">vs2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E9:59_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A14</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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">vs2</font></span>)</span> <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 class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E10:59_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E11:59_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A15</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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">vs2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59_1_1_2"><span class="lab"><font color="Green" title="E63">A14</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E12:59_1_1_2"/>
<font color="Maroon" title="c10">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="c3">vs2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59_1_1_2"><span class="lab"><font color="Green" title="E63">A14</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E13:59_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <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="c2">vs1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59_1_1_2"><span class="lab"><font color="Green" title="E62">A13</font></span></a>, <a class="ref" href="graph_2.html#T15" target="_self" title="GRAPH_2:th.15">Th15</a></span>;<br/><span class="kw">hence </span><a NAME="E14:59_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1_1_2"><span class="lab"><font color="Green" title="E61">A12</font></span></a>, <a class="txt" href="#E11:59_1_1_2"><span class="lab"><font color="Green" title="E64">A15</font></span></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E8:59"/><span class="lab"><font color="Green" title="E58">A16</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs1</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="c4">c1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="graph_2.html#D6" target="_self" title="GRAPH_2:def.6">Def7</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E59">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> implies  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v1</font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v1</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Olive" title="b2">v2</font>,<font color="Olive">b<sub>3</sub></font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:59_2"/><span class="lab"><font color="Green" title="E60">A18</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">and </span><br/><a NAME="E2:59_2"/><span class="lab"><font color="Green" title="E61">A19</font></span>: 
<font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v1</font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v1</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Olive" title="b2">v2</font>,<font color="Olive">b<sub>3</sub></font> )</span><br/><a NAME="E3:59_2"/><span class="lab"><font color="Green" title="E62">A20</font></span>: 
<font color="Maroon" title="c8">n</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="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_2"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="txt" href="#E2:59_2"><span class="lab"><font color="Green" title="E61">A19</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:59_2_1"/>
( <font color="Maroon" title="c8">n</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="c4">c1</font> or  ex <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">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="c5">c2</font> &amp; <font color="Maroon" title="c8">n</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="c4">c1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2"><span class="lab"><font color="Green" title="E62">A20</font></span></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:59_2_1_1"/><span class="lab"><font color="Green" title="E63">A21</font></span>: 
<font color="Maroon" title="c8">n</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="c4">c1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v1</font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v1</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Olive" title="b2">v2</font>,<font color="Olive">b<sub>3</sub></font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">v2</font> = <font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">v1</font> = <font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">n</font>;<br/><a NAME="E2:59_2_1_1"/><span class="lab"><font color="Green" title="E64">A22</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_2_1_1"><span class="lab"><font color="Green" title="E63">A21</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E3:59_2_1_1"/><span class="lab"><font color="Green" title="E65">A23</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_2_1_1"><span class="lab"><font color="Green" title="E63">A21</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:59_2_1_1"/><span class="lab"><font color="Green" title="E66">A24</font></span>: 
<font color="Maroon" title="c8">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="c4">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_2_1_1"><span class="lab"><font color="Green" title="E63">A21</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:59_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E6:59_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A25</font></span>: 
<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="graph_2.html#D6" target="_self" title="GRAPH_2:def.6">Def7</a></span>;<br/><a NAME="E7:59_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A26</font></span>: 
<font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E8:59_2_1_1"/><span class="lab"><font color="Green" title="E69">A27</font></span>: 
<font color="Maroon" title="c8">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">vs1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E4:59_2_1_1"><span class="lab"><font color="Green" title="E66">A24</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E9:59_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A28</font></span>: 
<font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2_1_1"><span class="lab"><font color="Green" title="E65">A23</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><a NAME="E10:59_2_1_1"/><span class="lab"><font color="Green" title="E71">A29</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59_2_1_1"><span class="lab"><font color="Green" title="E67">A25</font></span></a>, <a class="ref" href="graph_2.html#T14" target="_self" title="GRAPH_2:th.14">Th14</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E11:59_2_1_1"/><span class="lab"><font color="Green" title="E72">A30</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2_1_1"><span class="lab"><font color="Green" title="E65">A23</font></span></a>, <a class="txt" href="#E8:59_2_1_1"><span class="lab"><font color="Green" title="E69">A27</font></span></a>, <a class="ref" href="graph_2.html#T14" target="_self" title="GRAPH_2:th.14">Th14</a></span>;<br/><a NAME="E12:59_2_1_1"/>
<font color="Maroon" title="c4">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E3:59_2_1_1"><span class="lab"><font color="Green" title="E65">A23</font></span></a>, <a class="txt" href="#E4:59_2_1_1"><span class="lab"><font color="Green" title="E66">A24</font></span></a>, <a class="ref" href="graph_2.html#D6" target="_self" title="GRAPH_2:def.6">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E13:59_2_1_1"/>
 ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Olive" title="b2">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_2_1_1"><span class="lab"><font color="Green" title="E64">A22</font></span></a>, <a class="txt" href="#E9:59_2_1_1"><span class="lab"><font color="Green" title="E70">A28</font></span></a>, <a class="txt" href="#E7:59_2_1_1"><span class="lab"><font color="Green" title="E68">A26</font></span></a>, <a class="txt" href="#E11:59_2_1_1"><span class="lab"><font color="Green" title="E72">A30</font></span></a>, <a class="txt" href="#E10:59_2_1_1"><span class="lab"><font color="Green" title="E71">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:59_2_1_2"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">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="c5">c2</font> &amp; <font color="Maroon" title="c8">n</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="c4">c1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v1</font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v1</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Olive" title="b2">v2</font>,<font color="Olive">b<sub>3</sub></font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:59_2_1_2"/><span class="lab"><font color="Green" title="E63">A31</font></span>: 
<font color="Maroon" title="c9">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="c5">c2</font>
 <span class="kw">and </span><br/><a NAME="E4:59_2_1_2"/><span class="lab"><font color="Green" title="E64">A32</font></span>: 
<font color="Maroon" title="c8">n</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="c4">c1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font>
 ;<br/><a NAME="E5:59_2_1_2"/><span class="lab"><font color="Green" title="E65">A33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2_1_2"><span class="lab"><font color="Green" title="E63">A31</font></span></a>, <a class="txt" href="#E4:59_2_1_2"><span class="lab"><font color="Green" title="E64">A32</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E6:59_2_1_2"/><span class="lab"><font color="Green" title="E66">A34</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E7:59_2_1_2"/><span class="lab"><font color="Green" title="E67">A35</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2_1_2"><span class="lab"><font color="Green" title="E63">A31</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E8:59_2_1_2"/><span class="lab"><font color="Green" title="E68">A36</font></span>: 
<font color="Maroon" title="c9">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="c5">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2_1_2"><span class="lab"><font color="Green" title="E63">A31</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E9:59_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A37</font></span>: 
<font color="Maroon" title="c9">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">vs2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">k</font> = <font color="Maroon" title="c9">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2_1_2"><span class="lab"><font color="Green" title="E63">A31</font></span></a></span>;<br/><a NAME="E11:59_2_1_2"/><span class="lab"><font color="Green" title="E70">A38</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2_1_2"><span class="lab"><font color="Green" title="E63">A31</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E12:59_2_1_2"/><span class="lab"><font color="Green" title="E71">A39</font></span>: 
<font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E8:59_2_1_2"><span class="lab"><font color="Green" title="E68">A36</font></span></a>, <a class="txt" href="#E6:59_2_1_2"><span class="lab"><font color="Green" title="E66">A34</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E13:59_2_1_2"/><span class="lab"><font color="Green" title="E72">A40</font></span>: 
<font color="Maroon" title="c10">k</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">vs2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E8:59_2_1_2"><span class="lab"><font color="Green" title="E68">A36</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E14:59_2_1_2"/>
<font color="Maroon" title="c10">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">vs2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E8:59_2_1_2"><span class="lab"><font color="Green" title="E68">A36</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E16:59_2_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">j</font>
 <span class="kw">and </span><br/><a NAME="E17:59_2_1_2"/><span class="lab"><font color="Green" title="E73">A41</font></span>: 
<font color="Maroon" title="c11">j</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">vs2</font>
 <span class="kw">and </span><br/><a NAME="E18:59_2_1_2"/><span class="lab"><font color="Green" title="E74">A42</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:59_2_1_2"><span class="lab"><font color="Green" title="E70">A38</font></span></a>, <a class="ref" href="graph_2.html#T1" target="_self" title="GRAPH_2:th.1">Th1</a></span>;<br/><a NAME="E19:59_2_1_2"/><span class="lab"><font color="Green" title="E75">A43</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:59_2_1_2_1_1"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59_2_1_2"><span class="lab"><font color="Green" title="E67">A35</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:59_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E76">A44</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:59_2_1_2_1_1_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:59_2_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E8:59"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E4:59_2_1_2"><span class="lab"><font color="Green" title="E64">A32</font></span></a>, <a class="txt" href="#E1:59_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A44</font></span></a>, <a class="ref" href="graph_2.html#T14" target="_self" title="GRAPH_2:th.14">Th14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:59_2_1_2_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:59_2_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A45</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:59_2_1_2"><span class="lab"><font color="Green" title="E74">A42</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <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="c2">vs1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E4:59_2_1_2"><span class="lab"><font color="Green" title="E64">A32</font></span></a>, <a class="txt" href="#E18:59_2_1_2"><span class="lab"><font color="Green" title="E74">A42</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">vs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:59_2_1_2"><span class="lab"><font color="Green" title="E73">A41</font></span></a>, <a class="txt" href="#E18:59_2_1_2"><span class="lab"><font color="Green" title="E74">A42</font></span></a>, <a class="txt" href="#E2:59_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A45</font></span></a>, <a class="ref" href="graph_2.html#T15" target="_self" title="GRAPH_2:th.15">Th15</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">set </span><font color="Maroon" title="c12">v2</font> = <font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c13">v1</font> = <font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font>;<br/><a NAME="E20:59_2_1_2"/><span class="lab"><font color="Green" title="E76">A46</font></span>: 
<font color="Maroon" title="c5">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E7:59_2_1_2"><span class="lab"><font color="Green" title="E67">A35</font></span></a>, <a class="txt" href="#E8:59_2_1_2"><span class="lab"><font color="Green" title="E68">A36</font></span></a>, <a class="ref" href="graph_2.html#D6" target="_self" title="GRAPH_2:def.6">Def7</a></span>;<br/><a NAME="E21:59_2_1_2"/><span class="lab"><font color="Green" title="E77">A47</font></span>: 
<font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59_2_1_2"><span class="lab"><font color="Green" title="E67">A35</font></span></a>, <a class="txt" href="#E9:59_2_1_2"><span class="lab"><font color="Green" title="E69">A37</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c4">c1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_2_1_2"><span class="lab"><font color="Green" title="E64">A32</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">vs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E7:59_2_1_2"><span class="lab"><font color="Green" title="E67">A35</font></span></a>, <a class="txt" href="#E13:59_2_1_2"><span class="lab"><font color="Green" title="E72">A40</font></span></a>, <a class="ref" href="graph_2.html#T15" target="_self" title="GRAPH_2:th.15">Th15</a></span>
;<br/><span class="kw">hence </span><a NAME="E23:59_2_1_2"/>
 ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Olive" title="b2">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_2_1_2"><span class="lab"><font color="Green" title="E65">A33</font></span></a>, <a class="txt" href="#E21:59_2_1_2"><span class="lab"><font color="Green" title="E77">A47</font></span></a>, <a class="txt" href="#E12:59_2_1_2"><span class="lab"><font color="Green" title="E71">A39</font></span></a>, <a class="txt" href="#E20:59_2_1_2"><span class="lab"><font color="Green" title="E76">A46</font></span></a>, <a class="txt" href="#E19:59_2_1_2"><span class="lab"><font color="Green" title="E75">A43</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E60">A48</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">vs2</font></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="c4">c1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">c2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E56">A5</font></span></a></span>
<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"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
;<br/>
<span class="kw">thus </span><a NAME="E11:59"/>
<font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font> is    <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_4"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="graph_1.html#D12" title="GRAPH_1:def.12">GRAPH_1:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive">b<sub>1</sub></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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; (  for <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Olive">b<sub>2</sub></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="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> ) ) &amp; (  for <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Olive">b<sub>2</sub></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="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> or  ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> ) ) ) )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:59_4_1"/><span class="lab"><font color="Green" title="E61">A49</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">and </span><br/><a NAME="E2:59_4_1"/><span class="lab"><font color="Green" title="E62">A50</font></span>: 
<font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font></span><br/><a NAME="E3:59_4_1"/><span class="lab"><font color="Green" title="E63">A51</font></span>: 
<font color="Maroon" title="c8">n</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="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_4_1"><span class="lab"><font color="Green" title="E61">A49</font></span></a>, <a class="txt" href="#E2:59_4_1"><span class="lab"><font color="Green" title="E62">A50</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_4_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:59_4_1_1"/>
( <font color="Maroon" title="c8">n</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="c4">c1</font> or  ex <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">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="c5">c2</font> &amp; <font color="Maroon" title="c8">n</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="c4">c1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_4_1"><span class="lab"><font color="Green" title="E63">A51</font></span></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:59_4_1_1_1"/><span class="lab"><font color="Green" title="E64">A52</font></span>: 
<font color="Maroon" title="c8">n</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="c4">c1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font></span><br/><div class="add"><a NAME="E2:59_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A53</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E3:59_4_1_1_1"/><span class="lab"><font color="Green" title="E66">A54</font></span>: 
<font color="Maroon" title="c8">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="c4">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_4_1_1_1"><span class="lab"><font color="Green" title="E64">A52</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:59_4_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_4_1_1_1"><span class="lab"><font color="Green" title="E64">A52</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">hence </span><a NAME="E5:59_4_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_4_1_1_1"><span class="lab"><font color="Green" title="E66">A54</font></span></a>, <a class="txt" href="#E2:59_4_1_1_1"><span class="lab"><font color="Green" title="E65">A53</font></span></a>, <a class="ref" href="graph_1.html#D12" title="GRAPH_1:def.12">GRAPH_1:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:59_4_1_1_2"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">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="c5">c2</font> &amp; <font color="Maroon" title="c8">n</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="c4">c1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:59_4_1_1_2"/><span class="lab"><font color="Green" title="E64">A55</font></span>: 
<font color="Maroon" title="c9">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="c5">c2</font>
 <span class="kw">and </span><br/><a NAME="E4:59_4_1_1_2"/><span class="lab"><font color="Green" title="E65">A56</font></span>: 
<font color="Maroon" title="c8">n</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="c4">c1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font>
 ;<br/><a NAME="E5:59_4_1_1_2"/><span class="lab"><font color="Green" title="E66">A57</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_4_1_1_2"><span class="lab"><font color="Green" title="E64">A55</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E6:59_4_1_1_2"/><span class="lab"><font color="Green" title="E67">A58</font></span>: 
<font color="Maroon" title="c9">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="c5">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_4_1_1_2"><span class="lab"><font color="Green" title="E64">A55</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E7:59_4_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_4_1_1_2"><span class="lab"><font color="Green" title="E64">A55</font></span></a>, <a class="txt" href="#E4:59_4_1_1_2"><span class="lab"><font color="Green" title="E65">A56</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E8:59_4_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_4_1_1_2"><span class="lab"><font color="Green" title="E64">A55</font></span></a>, <a class="txt" href="#E5:59_4_1_1_2"><span class="lab"><font color="Green" title="E66">A57</font></span></a>, <a class="txt" href="#E6:59_4_1_1_2"><span class="lab"><font color="Green" title="E67">A58</font></span></a>, <a class="ref" href="graph_1.html#D12" title="GRAPH_1:def.12">GRAPH_1:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:59_4"/>
 ex <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive">b<sub>1</sub></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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; (  for <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Olive">b<sub>2</sub></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="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> ) ) &amp; (  for <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Olive">b<sub>2</sub></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="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> or  ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E60">A48</font></span></a>, <a class="txt" href="#E7:59"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="txt" href="#E9:59"><span class="lab"><font color="Green" title="E59">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
