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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="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">sc</font> being   <a href="graph_2.html#V3" title="GRAPH_2:attr.3">simple</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font>  st 2 <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="Olive" title="b3">sc</font> &amp; <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">sc</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="b3">sc</font> holds <br/><font color="Olive" title="b1">vs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">vs2</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">sc</font> being   <a href="graph_2.html#V3" title="GRAPH_2:attr.3">simple</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font>  st 2 <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="Olive" title="b1">sc</font> &amp; <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">sc</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="b1">sc</font> holds <br/><font color="Maroon" title="c2">vs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">sc</font> be   <a href="graph_2.html#V3" title="GRAPH_2:attr.3">simple</a>   <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"> ( 2 <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="c4">sc</font> &amp; <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">sc</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="c4">sc</font> implies <font color="Maroon" title="c2">vs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:65"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
2 <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="c4">sc</font>
 <span class="kw">and </span><br/><a NAME="E2:65"/><span class="lab"><font color="Green" title="E57">A2</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">sc</font>
 <span class="kw">and </span><br/><a NAME="E3:65"/><span class="lab"><font color="Green" title="E58">A3</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="c4">sc</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font></span><br/>

<a NAME="E4:65"/><span class="lab"><font color="Green" title="E59">A4</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">sc</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="#E2:65"><span class="lab"><font color="Green" title="E57">A2</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">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $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="c2">vs1</font> &amp; <font color="Maroon" title="c2">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">vs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/>
<span class="kw">set </span><font color="Maroon" title="c5">TG</font> =  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">G</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">SG</font> =  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">G</font>;<br/>
<a NAME="E5:65"/><span class="lab"><font color="Green" title="E60">A5</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:65"/><span class="lab"><font color="Green" title="E61">A6</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">vs2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E7:65"/><span class="lab"><font color="Green" title="E62">A7</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="c4">sc</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="#E3:65"><span class="lab"><font color="Green" title="E58">A3</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">assume </span><a NAME="E8:65"/><span class="lab"><font color="Green" title="E63">A8</font></span>: 
<font color="Maroon" title="c2">vs1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">vs2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E9:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A9</font></span>: 
 ex <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E7:65"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">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="E11:65"/><span class="lab"><font color="Green" title="E65">A10</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>]
 <span class="kw">and </span><br/><a NAME="E12:65"/><span class="lab"><font color="Green" title="E66">A11</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E9:65"><span class="lab"><font color="Green" title="E64">A9</font></span></a>);<br/></span>
<a NAME="E13:65"/><span class="lab"><font color="Green" title="E67">A12</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:65"><span class="lab"><font color="Green" title="E65">A10</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="65_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="E1:65_1"/>
( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:65"><span class="lab"><font color="Green" title="E67">A12</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="65_1_1"><span class="kw">suppose </span></a><a NAME="E1:65_1_1"/><span class="lab"><font color="Green" title="E68">A13</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c8">v23</font> = <font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <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> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">v22</font> = <font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <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">v21</font> = <font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1;<br/><span class="kw">set </span><font color="Maroon" title="c11">v13</font> = <font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <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> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">v12</font> = <font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <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">v11</font> = <font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1;<br/><a NAME="E2:65_1_1"/><span class="lab"><font color="Green" title="E69">A14</font></span>: 
<span class="p1">(<span class="default">1 <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> 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="#E1:65"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E3:65_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A15</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"><span class="p2">(<span class="default">1 <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> 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"><span class="p2">(<span class="default">1 <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> 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></span>;<br/><a NAME="E4:65_1_1"/><span class="lab"><font color="Green" title="E71">A16</font></span>: 
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="#E2:65_1_1"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:65_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A17</font></span>: 
<font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <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> 1
 <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></span>;<br/><a NAME="E6:65_1_1"/><span class="lab"><font color="Green" title="E73">A18</font></span>: 
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="c4">sc</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:65_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A19</font></span>: 
<font color="Maroon" title="c4">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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> 1,<font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <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="#E3:65"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="graph_2.html#D6" target="_self" title="GRAPH_2:def.6">Def7</a></span>;<br/><a NAME="E8:65_1_1"/>
<font color="Maroon" title="c4">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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> 1,<font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <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="#E2:65"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="txt" href="#E6:65_1_1"><span class="lab"><font color="Green" title="E73">A18</font></span></a>, <a class="ref" href="graph_2.html#D6" target="_self" title="GRAPH_2:def.6">Def7</a></span>;<br/><a NAME="E9:65_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A20</font></span>: 
( ( <font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:65_1_1"><span class="lab"><font color="Green" title="E74">A19</font></span></a>, <a class="ref" href="graph_2.html#T30" target="_self" title="GRAPH_2:th.30">Th32</a></span>;<br/><a NAME="E10:65_1_1"/><span class="lab"><font color="Green" title="E76">A21</font></span>: 
<font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E7:65"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="txt" href="#E4:65_1_1"><span class="lab"><font color="Green" title="E71">A16</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">vs</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><span class="kw"> such that </span><br/><a NAME="E12:65_1_1"/><span class="lab"><font color="Green" title="E77">A22</font></span>: 
<font color="Maroon" title="c14">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c4">sc</font>
 <span class="kw">and </span><br/><a NAME="E13:65_1_1"/><span class="lab"><font color="Green" title="E78">A23</font></span>: 
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">vs</font> &amp; <font color="Maroon" title="c14">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">vs</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#D9" target="_self" title="GRAPH_2:def.9">Def10</a></span>;<br/><a NAME="E14:65_1_1"/>
<font color="Maroon" title="c4">sc</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E56">A1</font></span></a></span>;<br/><a NAME="E15:65_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A24</font></span>: 
( <font color="Maroon" title="c14">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs1</font> or <font color="Maroon" title="c14">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="#E8:65"><span class="lab"><font color="Green" title="E63">A8</font></span></a>, <a class="txt" href="#E12:65_1_1"><span class="lab"><font color="Green" title="E77">A22</font></span></a>, <a class="ref" href="graph_2.html#T34" target="_self" title="GRAPH_2:th.34">Th37</a></span>;<br/><a NAME="E16:65_1_1"/><span class="lab"><font color="Green" title="E80">A25</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"><span class="p2">(<span class="default">1 <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> 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"><span class="p2">(<span class="default">1 <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> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E7:65"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="txt" href="#E2:65_1_1"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><a NAME="E17:65_1_1"/><span class="lab"><font color="Green" title="E81">A26</font></span>: 
<font color="Maroon" title="c4">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <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> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <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> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="graph_2.html#D6" target="_self" title="GRAPH_2:def.6">Def7</a></span>;<br/><a NAME="E18:65_1_1"/><span class="lab"><font color="Green" title="E82">A27</font></span>: 
<font color="Maroon" title="c4">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <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> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <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> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E57">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="E19:65_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A28</font></span>: 
<font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <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> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:65"><span class="lab"><font color="Green" title="E65">A10</font></span></a>, <a class="txt" href="#E1:65_1_1"><span class="lab"><font color="Green" title="E68">A13</font></span></a>, <a class="txt" href="#E5:65_1_1"><span class="lab"><font color="Green" title="E72">A17</font></span></a>, <a class="txt" href="#E10:65_1_1"><span class="lab"><font color="Green" title="E76">A21</font></span></a>, <a class="txt" href="#E17:65_1_1"><span class="lab"><font color="Green" title="E81">A26</font></span></a>, <a class="txt" href="#E9:65_1_1"><span class="lab"><font color="Green" title="E75">A20</font></span></a>, <a class="ref" href="graph_2.html#T30" target="_self" title="GRAPH_2:th.30">Th32</a></span>;<br/><a NAME="E20:65_1_1"/>
<font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <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> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:65"><span class="lab"><font color="Green" title="E65">A10</font></span></a>, <a class="txt" href="#E1:65_1_1"><span class="lab"><font color="Green" title="E68">A13</font></span></a>, <a class="txt" href="#E5:65_1_1"><span class="lab"><font color="Green" title="E72">A17</font></span></a>, <a class="txt" href="#E10:65_1_1"><span class="lab"><font color="Green" title="E76">A21</font></span></a>, <a class="txt" href="#E18:65_1_1"><span class="lab"><font color="Green" title="E82">A27</font></span></a>, <a class="txt" href="#E17:65_1_1"><span class="lab"><font color="Green" title="E81">A26</font></span></a>, <a class="txt" href="#E9:65_1_1"><span class="lab"><font color="Green" title="E75">A20</font></span></a>, <a class="ref" href="graph_2.html#T30" target="_self" title="GRAPH_2:th.30">Th32</a></span>;<br/><a NAME="E21:65_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">1 <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> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">vs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E7:65"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="txt" href="#E2:65_1_1"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="txt" href="#E5:65_1_1"><span class="lab"><font color="Green" title="E72">A17</font></span></a>, <a class="txt" href="#E3:65_1_1"><span class="lab"><font color="Green" title="E70">A15</font></span></a>, <a class="txt" href="#E10:65_1_1"><span class="lab"><font color="Green" title="E76">A21</font></span></a>, <a class="txt" href="#E16:65_1_1"><span class="lab"><font color="Green" title="E80">A25</font></span></a>, <a class="txt" href="#E19:65_1_1"><span class="lab"><font color="Green" title="E83">A28</font></span></a>, <a class="txt" href="#E13:65_1_1"><span class="lab"><font color="Green" title="E78">A23</font></span></a>, <a class="txt" href="#E15:65_1_1"><span class="lab"><font color="Green" title="E79">A24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E22:65_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E7:65"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="txt" href="#E15:65_1_1"><span class="lab"><font color="Green" title="E79">A24</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="65_1_2"><span class="kw">suppose </span></a><a NAME="E1:65_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:65_1_2"/><span class="kw">then </span>
1 <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="c7">k</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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">k1</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="E4:65_1_2"/><span class="lab"><font color="Green" title="E68">A29</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k1</font>
 <span class="kw">and </span><br/><a NAME="E5:65_1_2"/><span class="lab"><font color="Green" title="E69">A30</font></span>: 
<font color="Maroon" title="c8">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">and </span><br/><a NAME="E6:65_1_2"/><span class="lab"><font color="Green" title="E70">A31</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#T1" target="_self" title="GRAPH_2:th.1">Th1</a></span>;<br/><a NAME="E7:65_1_2"/><span class="lab"><font color="Green" title="E71">A32</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:65"><span class="lab"><font color="Green" title="E65">A10</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E8:65_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A33</font></span>: 
<font color="Maroon" title="c8">k1</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="#E5:65_1_2"><span class="lab"><font color="Green" title="E69">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:65_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A34</font></span>: 
<font color="Maroon" title="c8">k1</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="#E4:65_1_2"><span class="lab"><font color="Green" title="E68">A29</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E10:65_1_2"/><span class="lab"><font color="Green" title="E74">A35</font></span>: 
<font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k1</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">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65_1_2"><span class="lab"><font color="Green" title="E68">A29</font></span></a>, <a class="txt" href="#E8:65_1_2"><span class="lab"><font color="Green" title="E72">A33</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><a NAME="E11:65_1_2"/><span class="lab"><font color="Green" title="E75">A36</font></span>: 
<font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">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="c7">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E7:65"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="txt" href="#E5:65"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="txt" href="#E6:65"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="txt" href="#E11:65"><span class="lab"><font color="Green" title="E65">A10</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E12:65_1_2"/><span class="lab"><font color="Green" title="E76">A37</font></span>: 
<font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</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="c7">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:65"><span class="lab"><font color="Green" title="E65">A10</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E13:65_1_2"/><span class="lab"><font color="Green" title="E77">A38</font></span>: 
<font color="Maroon" title="c8">k1</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">sc</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E6:65_1_2"><span class="lab"><font color="Green" title="E70">A31</font></span></a>, <a class="txt" href="#E7:65_1_2"><span class="lab"><font color="Green" title="E71">A32</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E14:65_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k1</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">k1</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">k1</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="#E2:65"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="txt" href="#E4:65_1_2"><span class="lab"><font color="Green" title="E68">A29</font></span></a>, <a class="ref" href="graph_2.html#D6" target="_self" title="GRAPH_2:def.6">Def7</a></span>;<br/><a NAME="E15:65_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A39</font></span>: 
( (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k1</font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font> ) or (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">vs1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:65_1_2"><span class="lab"><font color="Green" title="E70">A31</font></span></a>, <a class="ref" href="graph_1.html#D10" title="GRAPH_1:def.10">GRAPH_1:def 10</a></span>;<br/><a NAME="E16:65_1_2"/>
<font color="Maroon" title="c4">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k1</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="c8">k1</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="c8">k1</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="#E3:65"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="#E4:65_1_2"><span class="lab"><font color="Green" title="E68">A29</font></span></a>, <a class="txt" href="#E13:65_1_2"><span class="lab"><font color="Green" title="E77">A38</font></span></a>, <a class="ref" href="graph_2.html#D6" target="_self" title="GRAPH_2:def.6">Def7</a></span>;<br/><a NAME="E17:65_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A40</font></span>: 
( (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k1</font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font> ) or (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:65_1_2"><span class="lab"><font color="Green" title="E70">A31</font></span></a>, <a class="ref" href="graph_1.html#D10" title="GRAPH_1:def.10">GRAPH_1:def 10</a></span>;<br/><a NAME="E18:65_1_2"/>
<font color="Maroon" title="c3">vs2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k1</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="c8">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E7:65"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="txt" href="#E4:65_1_2"><span class="lab"><font color="Green" title="E68">A29</font></span></a>, <a class="txt" href="#E8:65_1_2"><span class="lab"><font color="Green" title="E72">A33</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><span class="kw">hence </span><a NAME="E19:65_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:65"><span class="lab"><font color="Green" title="E65">A10</font></span></a>, <a class="txt" href="#E12:65"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="txt" href="#E5:65_1_2"><span class="lab"><font color="Green" title="E69">A30</font></span></a>, <a class="txt" href="#E9:65_1_2"><span class="lab"><font color="Green" title="E73">A34</font></span></a>, <a class="txt" href="#E10:65_1_2"><span class="lab"><font color="Green" title="E74">A35</font></span></a>, <a class="txt" href="#E12:65_1_2"><span class="lab"><font color="Green" title="E76">A37</font></span></a>, <a class="txt" href="#E11:65_1_2"><span class="lab"><font color="Green" title="E75">A36</font></span></a>, <a class="txt" href="#E15:65_1_2"><span class="lab"><font color="Green" title="E78">A39</font></span></a>, <a class="txt" href="#E17:65_1_2"><span class="lab"><font color="Green" title="E79">A40</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>
