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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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">c</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">c</font> <a href="graph_2.html#R2" title="GRAPH_2:pred.2">alternates_vertices_in</a> <font color="Maroon" title="c1">G</font> holds <br/> ex <font color="Olive" title="b2">vs1</font>, <font color="Olive" title="b3">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> st <br/>( <font color="Olive" title="b2">vs1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">vs2</font> &amp; <font color="Olive" title="b2">vs1</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b3">vs2</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Olive" title="b1">c</font> &amp; ( for <font color="Olive" title="b4">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>  holds <br/>(  not <font color="Olive" title="b4">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Olive" title="b1">c</font> or <font color="Olive" title="b4">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">vs1</font> or <font color="Olive" title="b4">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">vs2</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">c</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">c</font> <a href="graph_2.html#R2" title="GRAPH_2:pred.2">alternates_vertices_in</a> <font color="Maroon" title="c1">G</font> implies  ex <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> st <br/>( <font color="Olive" title="b1">vs1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">vs2</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="Maroon" title="c2">c</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="Maroon" title="c2">c</font> &amp; ( for <font color="Olive" title="b3">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>  holds <br/>(  not <font color="Olive" title="b3">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font> or <font color="Olive" title="b3">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">vs1</font> or <font color="Olive" title="b3">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">vs2</font> ) ) ) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">X</font> = 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="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">Y</font> = 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="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">p1</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="E2:55"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<font color="Maroon" title="c5">p1</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#T36" target="_self" title="GRAPH_2:th.36">Th36</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:55"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
<font color="Maroon" title="c2">c</font> <a href="graph_2.html#R2" title="GRAPH_2:pred.2">alternates_vertices_in</a> <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <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> st <br/>( <font color="Olive" title="b1">vs1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">vs2</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="Maroon" title="c2">c</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="Maroon" title="c2">c</font> &amp; ( for <font color="Olive" title="b3">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>  holds <br/>(  not <font color="Olive" title="b3">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font> or <font color="Olive" title="b3">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">vs1</font> or <font color="Olive" title="b3">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">vs2</font> ) ) )</span><br/>

<a NAME="E4:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A3</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">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#D8" target="_self" title="GRAPH_2:def.8">Def8</a></span>;<br/>
<a NAME="E5:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A4</font></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">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E6:55"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
;<br/>
<a NAME="E7:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A6</font></span>: 
<font color="Maroon" title="c5">p1</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="c5">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="ref" href="graph_2.html#T38" target="_self" title="GRAPH_2:th.38">Th38</a></span>;<br/>
<a NAME="E8:55"/><span class="lab"><font color="Green" title="E52">A7</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default">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="p3">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default">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="p3">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="graph_2.html#T39" target="_self" title="GRAPH_2:th.39">Th39</a></span>;<br/>
<a NAME="E9:55"/><span class="lab"><font color="Green" title="E53">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c</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="#E2:55"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="graph_2.html#D7" target="_self" title="GRAPH_2:def.7">Def7</a></span>;<br/>
<a NAME="E10:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:55"><span class="lab"><font color="Green" title="E48">A3</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">then </span><span class="kw">consider </span><font color="Maroon" title="c6">p2</font> being   <a href="graph_2.html#V1" title="GRAPH_2:attr.1">TwoValued</a>   <a href="graph_2.html#V2" title="GRAPH_2:attr.2">Alternating</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E12:55"/><span class="lab"><font color="Green" title="E55">A10</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">and </span><br/><a NAME="E13:55"/><span class="lab"><font color="Green" title="E56">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p2</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="c5">p1</font>
 <span class="kw">and </span><br/><a NAME="E14:55"/><span class="lab"><font color="Green" title="E57">A12</font></span>: 
<font color="Maroon" title="c6">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:55"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="graph_2.html#T23" target="_self" title="GRAPH_2:th.23">Th23</a></span>;<br/>
<a NAME="E15:55"/><span class="lab"><font color="Green" title="E58">A13</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p1</font> <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="c6">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:55"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<a NAME="E16:55"/>
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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:55"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E17:55"/><span class="kw">then </span>
2 <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">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E18:55"/><span class="kw">then </span>
<font color="Maroon" title="c5">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E19:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A14</font></span>: 
( <font color="Maroon" title="c5">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> or <font color="Maroon" title="c5">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:55"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E20:55"/>
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="c5">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:55"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E21:55"/><span class="kw">then </span>
<font color="Maroon" title="c5">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E22:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A15</font></span>: 
( <font color="Maroon" title="c5">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> or <font color="Maroon" title="c5">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:55"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">p2</font> = <font color="Maroon" title="c6">p2</font> as    <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">by</span> <span class="lab"><a class="txt" href="#E3:55"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E8:55"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E6:55"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E12:55"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E19:55"><span class="lab"><font color="Green" title="E59">A14</font></span></a>, <a class="ref" href="graph_2.html#T38" target="_self" title="GRAPH_2:th.38">Th38</a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">p1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c5">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">vs2</font> &amp; <font color="Maroon" title="c5">p1</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Olive" title="b1">vs2</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font> &amp; ( for <font color="Olive" title="b2">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>  holds <br/>(  not <font color="Olive" title="b2">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font> or <font color="Olive" title="b2">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> or <font color="Olive" title="b2">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">vs2</font> ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c7">p2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">p2</font> &amp; <font color="Maroon" title="c5">p1</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font> &amp; ( for <font color="Olive" title="b1">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>  holds <br/>(  not <font color="Olive" title="b1">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font> or <font color="Olive" title="b1">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> or <font color="Olive" title="b1">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E24:55"/>
<font color="Maroon" title="c5">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E6:55"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E14:55"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="ref" href="graph_2.html#T38" target="_self" title="GRAPH_2:th.38">Th38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p1</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font> &amp; ( for <font color="Olive" title="b1">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>  holds <br/>(  not <font color="Olive" title="b1">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font> or <font color="Olive" title="b1">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> or <font color="Olive" title="b1">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E25:55"/>
<font color="Maroon" title="c5">p1</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E46">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">p2</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font> &amp; ( for <font color="Olive" title="b1">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>  holds <br/>(  not <font color="Olive" title="b1">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font> or <font color="Olive" title="b1">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> or <font color="Olive" title="b1">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> ) ) )</span><br/>

<a NAME="E26:55"/><span class="lab"><font color="Green" title="E61">A16</font></span>: 
<font color="Maroon" title="c5">p1</font> is   <a href="graph_2.html#V1" title="GRAPH_2:attr.1">TwoValued</a>   <a href="graph_2.html#V2" title="GRAPH_2:attr.2">Alternating</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="graph_2.html#T40" target="_self" title="GRAPH_2:th.40">Th40</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><span class="kw">now </span></a><div class="add"><span class="kw">thus </span><a NAME="E1:55_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c</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="#E2:55"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E13:55"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="ref" href="graph_2.html#D7" target="_self" title="GRAPH_2:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">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">c</font> holds <br/><font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Maroon" title="c7">p2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c7">p2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><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="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="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> <font color="Maroon" title="c2">c</font> implies <font color="Maroon" title="c2">c</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="c7">p2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c7">p2</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><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:55_1"/><span class="lab"><font color="Green" title="E62">A17</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="E3:55_1"/><span class="lab"><font color="Green" title="E63">A18</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">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">c</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="c7">p2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c7">p2</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><br/><a NAME="E4:55_1"/><span class="lab"><font color="Green" title="E64">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> <font color="Maroon" title="c5">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:55"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="#E3:55_1"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E5:55_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A20</font></span>: 
<font color="Maroon" title="c7">p2</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="c7">p2</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="#E13:55"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="txt" href="#E2:55_1"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><a NAME="E6:55_1"/><span class="lab"><font color="Green" title="E66">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="c5">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="txt" href="#E4:55_1"><span class="lab"><font color="Green" title="E64">A19</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E7:55_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">p2</font> <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> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default">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="p3">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default">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="p3">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E6:55"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E12:55"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E15:55"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="txt" href="#E22:55"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="txt" href="#E19:55"><span class="lab"><font color="Green" title="E59">A14</font></span></a>, <a class="ref" href="graph_2.html#T38" target="_self" title="GRAPH_2:th.38">Th38</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E8:55_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A22</font></span>: 
( <font color="Maroon" title="c7">p2</font> <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> 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="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> or <font color="Maroon" title="c7">p2</font> <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> 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="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c5">p1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">n</font>;<br/><span class="kw">set </span><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c5">p1</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/><a NAME="E9:55_1"/><span class="lab"><font color="Green" title="E68">A23</font></span>: 
<font color="Maroon" title="c2">c</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="c5">p1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c5">p1</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="#E2:55"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E2:55_1"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="txt" href="#E3:55_1"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="ref" href="graph_2.html#D7" target="_self" title="GRAPH_2:def.7">Def7</a></span>;<br/><a NAME="E10:55_1"/><span class="lab"><font color="Green" title="E69">A24</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="c5">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:55"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="#E3:55_1"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E11:55_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A25</font></span>: 
<font color="Maroon" title="c7">p2</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="c7">p2</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="#E13:55"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E12:55_1"/><span class="lab"><font color="Green" title="E71">A26</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E13:55_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A27</font></span>: 
<font color="Maroon" title="c8">n</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="c5">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:55_1"><span class="lab"><font color="Green" title="E69">A24</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E14:55_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">p2</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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default">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="p3">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default">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="p3">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E6:55"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E12:55"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E15:55"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="txt" href="#E22:55"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="txt" href="#E19:55"><span class="lab"><font color="Green" title="E59">A14</font></span></a>, <a class="ref" href="graph_2.html#T38" target="_self" title="GRAPH_2:th.38">Th38</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E15:55_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A28</font></span>: 
( <font color="Maroon" title="c7">p2</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> or <font color="Maroon" title="c7">p2</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E16:55_1"/>
<font color="Maroon" title="c5">p1</font> <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> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default">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="p3">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default">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="p3">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:55"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E6:55_1"><span class="lab"><font color="Green" title="E66">A21</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E17:55_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">p1</font> <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> 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="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> or <font color="Maroon" title="c5">p1</font> <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> 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="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E18:55_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A29</font></span>: 
<font color="Maroon" title="c5">p1</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="c7">p2</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="#E8:55"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E26:55"><span class="lab"><font color="Green" title="E61">A16</font></span></a>, <a class="txt" href="#E7:55"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E12:55"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E13:55"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="txt" href="#E14:55"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="txt" href="#E22:55"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="txt" href="#E19:55"><span class="lab"><font color="Green" title="E59">A14</font></span></a>, <a class="txt" href="#E2:55_1"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="txt" href="#E4:55_1"><span class="lab"><font color="Green" title="E64">A19</font></span></a>, <a class="txt" href="#E10:55_1"><span class="lab"><font color="Green" title="E69">A24</font></span></a>, <a class="txt" href="#E8:55_1"><span class="lab"><font color="Green" title="E67">A22</font></span></a>, <a class="txt" href="#E15:55_1"><span class="lab"><font color="Green" title="E73">A28</font></span></a>, <a class="ref" href="graph_2.html#D4" target="_self" title="GRAPH_2:def.4">Def4</a>, <a class="ref" href="graph_2.html#T21" target="_self" title="GRAPH_2:th.21">Th21</a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><a NAME="E19:55_1"/>
<font color="Maroon" title="c5">p1</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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default">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="p3">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default">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="p3">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:55"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E13:55_1"><span class="lab"><font color="Green" title="E72">A27</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E20:55_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">p1</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> or <font color="Maroon" title="c5">p1</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E21:55_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">p1</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="c7">p2</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="#E8:55"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E26:55"><span class="lab"><font color="Green" title="E61">A16</font></span></a>, <a class="txt" href="#E7:55"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E12:55"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E13:55"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="txt" href="#E14:55"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="txt" href="#E22:55"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="txt" href="#E19:55"><span class="lab"><font color="Green" title="E59">A14</font></span></a>, <a class="txt" href="#E2:55_1"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="txt" href="#E12:55_1"><span class="lab"><font color="Green" title="E71">A26</font></span></a>, <a class="txt" href="#E10:55_1"><span class="lab"><font color="Green" title="E69">A24</font></span></a>, <a class="txt" href="#E8:55_1"><span class="lab"><font color="Green" title="E67">A22</font></span></a>, <a class="txt" href="#E15:55_1"><span class="lab"><font color="Green" title="E73">A28</font></span></a>, <a class="ref" href="graph_2.html#D4" target="_self" title="GRAPH_2:def.4">Def4</a>, <a class="ref" href="graph_2.html#T21" target="_self" title="GRAPH_2:th.21">Th21</a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><span class="kw">hence </span><a NAME="E22:55_1"/>
<font color="Maroon" title="c2">c</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="c7">p2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c7">p2</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="#E9:55_1"><span class="lab"><font color="Green" title="E68">A23</font></span></a>, <a class="txt" href="#E18:55_1"><span class="lab"><font color="Green" title="E74">A29</font></span></a>, <a class="txt" href="#E5:55_1"><span class="lab"><font color="Green" title="E65">A20</font></span></a>, <a class="txt" href="#E11:55_1"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="ref" href="graph_2.html#T31" target="_self" title="GRAPH_2:th.31">Th31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E28:55"/>
<font color="Maroon" title="c7">p2</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#D7" target="_self" title="GRAPH_2:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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>  holds <br/>(  not <font color="Olive" title="b1">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font> or <font color="Olive" title="b1">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> or <font color="Olive" title="b1">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">p</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="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">p</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font> or <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> or <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E29:55"/><span class="lab"><font color="Green" title="E62">A30</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> or <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">p9</font> = <font color="Maroon" title="c8">p</font> as   <a href="graph_2.html#V1" title="GRAPH_2:attr.1">TwoValued</a>   <a href="graph_2.html#V2" title="GRAPH_2:attr.2">Alternating</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="graph_2.html#T40" target="_self" title="GRAPH_2:th.40">Th40</a></span>;<br/>
<a NAME="E31:55"/><span class="lab"><font color="Green" title="E63">A31</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</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">c</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="#E29:55"><span class="lab"><font color="Green" title="E62">A30</font></span></a>, <a class="ref" href="graph_2.html#D7" target="_self" title="GRAPH_2:def.7">Def7</a></span>;<br/>
<a NAME="E32:55"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default">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="p3">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default">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="p3">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E29:55"><span class="lab"><font color="Green" title="E62">A30</font></span></a>, <a class="ref" href="graph_2.html#T39" target="_self" title="GRAPH_2:th.39">Th39</a></span>;<br/>
<a NAME="E33:55"/><span class="kw">then </span>
( <font color="Maroon" title="c9">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> or <font color="Maroon" title="c9">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:55"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="#E8:55"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E26:55"><span class="lab"><font color="Green" title="E61">A16</font></span></a>, <a class="txt" href="#E7:55"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E12:55"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E13:55"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="txt" href="#E14:55"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="txt" href="#E22:55"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="txt" href="#E19:55"><span class="lab"><font color="Green" title="E59">A14</font></span></a>, <a class="txt" href="#E31:55"><span class="lab"><font color="Green" title="E63">A31</font></span></a>, <a class="ref" href="graph_2.html#T22" target="_self" title="GRAPH_2:th.22">Th22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E34:55"/>
( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> or <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
