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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> ;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="graph_1.html#V3">oriented</a> <a href="graph_1.html#V7">finite</a> <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>6</sub></font>, <a href="graph_5.html#K8">Real&gt;=0</a> ;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>















<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>assume </b><a NAME="E1:100"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="graphsp.html#R6" target="_self">is_Input_of_Dijkstra_Alg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; 1 <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E2:100"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>6</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 holds <br/>ex <font color="Olive">b<sub>3</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ex <font color="Olive">b<sub>4</sub></font> being <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="graphsp.html#R4" target="_self">is_simple_vertex_seq_at</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) &amp; <font color="Olive">b<sub>4</sub></font> <a href="graphsp.html#R5" target="_self">is_oriented_edge_seq_of</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="graph_5.html#K10">cost</a> <font color="Olive">b<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> &amp; ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> implies <font color="Olive">b<sub>4</sub></font> <a href="graph_5.html#R9">islongestInShortestpath</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> ) )
 ;<br/>

<b>assume </b><a NAME="E3:100"/><i><font color="Green">E85</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 ) )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> =  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E4:100"/><i><font color="Green">E86</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E5:100"/><i><font color="Green">E87</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="ref" href="graphsp.html#D20" target="_self">Def20</a></i>;<br/>
<a NAME="E6:100"/><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graphsp.html#K2" target="_self">:=</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="graphsp.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E7:100"/><i><font color="Green">E89</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="graphsp.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E8:100"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="ref" href="graphsp.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E9:100"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="graphsp.html#K19" target="_self">Relax</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="graphsp.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E10:100"/><i><font color="Green">E91</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="ref" href="graphsp.html#D20" target="_self">Def20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>6</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:100"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="graph_5.html#R5">is_weight&gt;=0of</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="graph_5.html#D13">GRAPH_5:def 13</a></i>;<br/>
<a NAME="E13:100"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="graph_5.html#R6">is_weight_of</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="graph_5.html#T50">GRAPH_5:50</a></i>;<br/>
<a NAME="E14:100"/><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E15:100"/><i><font color="Green">E95</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="ref" href="graphsp.html#T42" target="_self">Th42</a></i>;<br/>
<a NAME="E16:100"/><i><font color="Green">E96</font></i>: 
(  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 )
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="ref" href="graphsp.html#T30" target="_self">Th30</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> =  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E17:100"/><i><font color="Green">E97</font></i>: 
(  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; not  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="ref" href="graphsp.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E18:100"/><b>then </b><i><font color="Green">E98</font></i>: 
 <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E19:100"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="graphsp.html#E52"><i><font color="Green">Lemma47</font></i></a></i>;<br/>
<a NAME="E20:100"/><i><font color="Green">E100</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="ref" href="graphsp.html#T42" target="_self">Th42</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> =  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="graphsp.html#E10:100"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graphsp.html#E16:100"><i><font color="Green">E96</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>19</sub></font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E23:100"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="graphsp.html#R4" target="_self">is_simple_vertex_seq_at</a> <font color="Maroon">c<sub>4</sub></font>, <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>19</sub></font> <a href="graphsp.html#R5" target="_self">is_oriented_edge_seq_of</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; ( not <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>19</sub></font> <a href="graph_5.html#R9">islongestInShortestpath</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> ) )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E2:100"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graphsp.html#E16:100"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graphsp.html#E17:100"><i><font color="Green">E97</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E25:100"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="ref" href="graphsp.html#D10" target="_self">Def10</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E26:100"/><i><font color="Green">E103</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="graphsp.html#E16:100"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graphsp.html#E82"><i><font color="Green">Lemma72</font></i></a></i>;<br/>
<a NAME="E27:100"/><b>then </b><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E5:100"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E15:100"><i><font color="Green">E95</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E28:100"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="graphsp.html#R2" target="_self">equal_at</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="ref" href="graphsp.html#T47" target="_self">Th47</a></i>;<br/>
<a NAME="E29:100"/>
<font color="Maroon">c<sub>19</sub></font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E23:100"><i><font color="Green">E101</font></i></a>, <a class="ref" href="graph_5.html#D18">GRAPH_5:def 18</a></i>;<br/>
<a NAME="E30:100"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="graph_5.html#R1">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="graph_5.html#D4">GRAPH_5:def 4</a></i>;<br/>
<a NAME="E31:100"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="graph_5.html#D3">GRAPH_5:def 3</a></i>;<br/>
<a NAME="E32:100"/><b>then </b><i><font color="Green">E107</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="graph_5.html#T8">GRAPH_5:8</a></i>;<br/>
<a NAME="E33:100"/><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E4:100"><i><font color="Green">E86</font></i></a>, <a class="txt" href="graphsp.html#E5:100"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E15:100"><i><font color="Green">E95</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E109</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E7:100"><i><font color="Green">E89</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E6:100"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graphsp.html#E16:100"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graphsp.html#E19:100"><i><font color="Green">E99</font></i></a>, <a class="txt" href="graphsp.html#E33:100"><i><font color="Green">E108</font></i></a>, <a class="ref" href="graphsp.html#T18" target="_self">Th18</a></i>
;<br/>
<i><font color="Green">E110</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E7:100"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graphsp.html#E27:100"><i><font color="Green">E104</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E6:100"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graphsp.html#E23:100"><i><font color="Green">E101</font></i></a>, <a class="txt" href="graphsp.html#E26:100"><i><font color="Green">E103</font></i></a>, <a class="txt" href="graphsp.html#E27:100"><i><font color="Green">E104</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self">Th19</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E36:100"/><i><font color="Green">E111</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="graphsp.html#E16:100"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graphsp.html#E83"><i><font color="Green">Lemma73</font></i></a></i>;<br/>
<a NAME="E37:100"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E16:100"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E38:100"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E39:100"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E5:100"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E15:100"><i><font color="Green">E95</font></i></a>, <a class="txt" href="graphsp.html#E36:100"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E40:100"/><b>then </b><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E36:100"><i><font color="Green">E111</font></i></a>, <a class="txt" href="graphsp.html#E38:100"><i><font color="Green">E112</font></i></a>, <a class="ref" href="graphsp.html#T32" target="_self">Th32</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_3"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">c<sub>23</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/><b>assume </b><a NAME="E1:100_3"/>
<span class="p1">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">c<sub>1</sub></font>, <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:100_3"/><b>then </b><i><font color="Green">E115</font></i>: 
( ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="graphsp.html#K18" target="_self">newpathcost</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="graphsp.html#D13" target="_self">Def13</a></i>;<br/><a NAME="E3:100_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E4:100_3"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E34:100"><i><font color="Green">E109</font></i></a>, <a class="txt" href="graphsp.html#E40:100"><i><font color="Green">E114</font></i></a>, <a class="txt" href="graphsp.html#E2:100_3"><i><font color="Green">E115</font></i></a>, <a class="ref" href="graphsp.html#T30" target="_self">Th30</a></i>;<br/></div><b>end;</b></div>
<a NAME="E42:100"/><b>then </b>
not <span class="p1">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>
<a NAME="E43:100"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E7:100"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graphsp.html#E9:100"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graphsp.html#E36:100"><i><font color="Green">E111</font></i></a>, <a class="txt" href="graphsp.html#E38:100"><i><font color="Green">E112</font></i></a>, <a class="txt" href="graphsp.html#E39:100"><i><font color="Green">E113</font></i></a>, <a class="txt" href="graphsp.html#E40:100"><i><font color="Green">E114</font></i></a>, <a class="ref" href="graphsp.html#D14" target="_self">Def14</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>23</sub></font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>24</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:100_4"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 )
 ;<br/><b>set </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font>;<br/><a NAME="E2:100_4"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100_4"><i><font color="Green">E116</font></i></a></i>;<br/><a NAME="E3:100_4"/><b>then </b><i><font color="Green">E117</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E10:100"><i><font color="Green">E91</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:100_4"/><b>then </b><i><font color="Green">E118</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graphsp.html#E83"><i><font color="Green">Lemma73</font></i></a></i>;<br/><a NAME="E5:100_4"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:100"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E15:100"><i><font color="Green">E95</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:100_4"/><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E7:100_4"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:100_4"><i><font color="Green">E117</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E8:100_4"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>set </b><font color="Maroon">c<sub>26</sub></font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font>;<br/><a NAME="E9:100_4"/><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E14:100"><i><font color="Green">E94</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4"><i><font color="Green">E117</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E10:100_4"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:100_4"><i><font color="Green">E117</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E11:100_4"/><b>then </b><i><font color="Green">E123</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:100_4"/><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E13:100_4"/><i><font color="Green">E125</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graphsp.html#E3:100_4"><i><font color="Green">E117</font></i></a>, <a class="txt" href="graphsp.html#E82"><i><font color="Green">Lemma72</font></i></a></i>;<br/><a NAME="E14:100_4"/><b>then </b><i><font color="Green">E126</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:100"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E15:100"><i><font color="Green">E95</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:100_4"/><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:100"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E7:100"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graphsp.html#E15:100"><i><font color="Green">E95</font></i></a>, <a class="txt" href="graphsp.html#E13:100_4"><i><font color="Green">E125</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:100_4"/><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E6:100"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graphsp.html#E16:100"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4"><i><font color="Green">E118</font></i></a>, <a class="txt" href="graphsp.html#E5:100_4"><i><font color="Green">E119</font></i></a>, <a class="txt" href="graphsp.html#E8:100_4"><i><font color="Green">E121</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E17:100_4"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E51"><i><font color="Green">Lemma46</font></i></a></i>;<br/><a NAME="E18:100_4"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E11:100_4"><i><font color="Green">E123</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E19:100_4"/><b>then </b><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E6:100"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graphsp.html#E16:100"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graphsp.html#E13:100_4"><i><font color="Green">E125</font></i></a>, <a class="txt" href="graphsp.html#E14:100_4"><i><font color="Green">E126</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E20:100_4"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E19:100"><i><font color="Green">E99</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4"><i><font color="Green">E117</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E21:100_4"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:100"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E15:100"><i><font color="Green">E95</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4"><i><font color="Green">E117</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E22:100_4"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:100"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E20:100"><i><font color="Green">E100</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4"><i><font color="Green">E117</font></i></a>, <a class="txt" href="graphsp.html#E20:100_4"><i><font color="Green">E130</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>set </b><font color="Maroon">c<sub>27</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font>;<br/><a NAME="E23:100_4"/><i><font color="Green">E133</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> &amp;  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graphsp.html#E16:100"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4"><i><font color="Green">E117</font></i></a>, <a class="txt" href="graphsp.html#E84"><i><font color="Green">Lemma74</font></i></a></i>;<br/><a NAME="E24:100_4"/><b>then </b><i><font color="Green">E134</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:100"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E15:100"><i><font color="Green">E95</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E25:100_4"/><i><font color="Green">E135</font></i>: 
( <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="graphsp.html#E16:100"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4"><i><font color="Green">E117</font></i></a>, <a class="txt" href="graphsp.html#E85"><i><font color="Green">Lemma75</font></i></a></i>;<br/><i><font color="Green">E136</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p5">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E7:100"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graphsp.html#E34:100"><i><font color="Green">E109</font></i></a>, <a class="txt" href="graphsp.html#E24:100_4"><i><font color="Green">E134</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p5">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E6:100"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graphsp.html#E23:100_4"><i><font color="Green">E133</font></i></a>, <a class="txt" href="graphsp.html#E24:100_4"><i><font color="Green">E134</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self">Th19</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p5">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E15:100"><i><font color="Green">E95</font></i></a>, <a class="txt" href="graphsp.html#E28:100"><i><font color="Green">E105</font></i></a>, <a class="txt" href="graphsp.html#E24:100_4"><i><font color="Green">E134</font></i></a>, <a class="txt" href="graphsp.html#E25:100_4"><i><font color="Green">E135</font></i></a>, <a class="ref" href="graphsp.html#D16" target="_self">Def16</a></i>
<br/>.= 
 <a href="graphsp.html#K13" target="_self">Weight</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4"><i><font color="Green">E116</font></i></a>, <a class="ref" href="graphsp.html#D20" target="_self">Def20</a></i>
;<br/><a NAME="E27:100_4"/><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E7:100"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graphsp.html#E14:100_4"><i><font color="Green">E126</font></i></a>, <a class="txt" href="graphsp.html#E19:100_4"><i><font color="Green">E129</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_4_2"/>
(  not <span class="p1">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font> or <span class="p1">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_1"><b>suppose </b></a><a NAME="E1:100_4_2_1"/><i><font color="Green">E138</font></i>: 
not <span class="p1">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:100_4_2_1"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E7:100"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graphsp.html#E9:100"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4"><i><font color="Green">E118</font></i></a>, <a class="txt" href="graphsp.html#E5:100_4"><i><font color="Green">E119</font></i></a>, <a class="txt" href="graphsp.html#E8:100_4"><i><font color="Green">E121</font></i></a>, <a class="txt" href="graphsp.html#E16:100_4"><i><font color="Green">E128</font></i></a>, <a class="ref" href="graphsp.html#D14" target="_self">Def14</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>28</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>29</sub></font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E4:100_4_2_1"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="graphsp.html#R4" target="_self">is_simple_vertex_seq_at</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>28</sub></font> holds <br/><font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>29</sub></font> <a href="graphsp.html#R5" target="_self">is_oriented_edge_seq_of</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>23</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> &amp; ( not <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>29</sub></font> <a href="graph_5.html#R9">islongestInShortestpath</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> ) )
 <b>by </b><i><a class="txt" href="graphsp.html#E2:100"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4"><i><font color="Green">E116</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>28</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>29</sub></font>;<br/><b>thus </b><a NAME="E5:100_4_2_1"/>
<font color="Maroon">c<sub>30</sub></font> <a href="graphsp.html#R4" target="_self">is_simple_vertex_seq_at</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E5:100"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E2:100_4_2_1"><i><font color="Green">E139</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4_2_1"><i><font color="Green">E140</font></i></a>, <a class="txt" href="graphsp.html#E90"><i><font color="Green">Lemma79</font></i></a></i>;<br/><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>32</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:100_4_2_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>30</sub></font> )
 ;<br/><a NAME="E2:100_4_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E4:100_4_2_1"><i><font color="Green">E140</font></i></a></i>;<br/><b>hence </b><a NAME="E3:100_4_2_1_1"/>
<font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E18:100"><i><font color="Green">E98</font></i></a></i>;<br/></div>
<b>end;</b></div><b>thus </b><a NAME="E7:100_4_2_1"/>
<font color="Maroon">c<sub>31</sub></font> <a href="graphsp.html#R5" target="_self">is_oriented_edge_seq_of</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E4:100_4_2_1"><i><font color="Green">E140</font></i></a></i>;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_4_2_1_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:100_4_2_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><div class="add"><a NAME="E2:100_4_2_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="graphsp.html#K19" target="_self">Relax</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E7:100"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4"><i><font color="Green">E117</font></i></a>, <a class="txt" href="graphsp.html#E21:100_4"><i><font color="Green">E131</font></i></a>, <a class="ref" href="graphsp.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E3:100_4_2_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graphsp.html#E9:100"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4"><i><font color="Green">E117</font></i></a>, <a class="txt" href="graphsp.html#E22:100_4"><i><font color="Green">E132</font></i></a></i>;<br/><a NAME="E4:100_4_2_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E5:100_4_2_1_2_1_1"/><b>then </b><i><font color="Green">E141</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="graphsp.html#E17:100"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_4_2_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>32</sub></font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>33</sub></font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>assume </b><a NAME="E1:100_4_2_1_2_1_1_1"/><i><font color="Green">E142</font></i>: 
( not <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>33</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 ;<br/><a NAME="E2:100_4_2_1_2_1_1_1"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>33</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E4:100_4_2_1_2_1_1_1"/><i><font color="Green">E144</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E10:100"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graphsp.html#E2:100_4_2_1_2_1_1_1"><i><font color="Green">E143</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:100_4_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E3:100"><i><font color="Green">E85</font></i></a>, <a class="txt" href="graphsp.html#E12:100"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_1_2_1_1_1"><i><font color="Green">E142</font></i></a>, <a class="txt" href="graphsp.html#E92"><i><font color="Green">Lemma81</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>35</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>36</sub></font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E7:100_4_2_1_2_1_1_1"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="graphsp.html#R4" target="_self">is_simple_vertex_seq_at</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>35</sub></font> holds <br/><font color="Maroon">c<sub>35</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>36</sub></font> <a href="graphsp.html#R5" target="_self">is_oriented_edge_seq_of</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>33</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> &amp; ( not <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>36</sub></font> <a href="graph_5.html#R9">islongestInShortestpath</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> ) )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E2:100"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_1_2_1_1_1"><i><font color="Green">E142</font></i></a></i>;<br/><a NAME="E8:100_4_2_1_2_1_1_1"/><i><font color="Green">E147</font></i>: 
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100_4_2_1_2_1_1_1"><i><font color="Green">E142</font></i></a>, <a class="txt" href="graphsp.html#E7:100_4_2_1_2_1_1_1"><i><font color="Green">E146</font></i></a>, <a class="ref" href="graphsp.html#T9" target="_self">Th9</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_4_2_1_2_1_1_1_1"/>
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="graphsp.html#E5:100_4_2_1_2_1_1"><i><font color="Green">E141</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:100_4_2_1_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>37</sub></font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E3:100_4_2_1_2_1_1_1_1_1"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">c<sub>37</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>23</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100_4"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_1_2_1_1_1"><i><font color="Green">E142</font></i></a>, <a class="txt" href="graphsp.html#E7:100_4_2_1_2_1_1_1"><i><font color="Green">E146</font></i></a>, <a class="ref" href="graph_5.html#D19">GRAPH_5:def 19</a></i>;<br/><b>thus </b><a NAME="E4:100_4_2_1_2_1_1_1_1_1"/>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E4:100_4_2_1"><i><font color="Green">E140</font></i></a>, <a class="txt" href="graphsp.html#E8:100_4_2_1_2_1_1_1"><i><font color="Green">E147</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4_2_1_2_1_1_1_1_1"><i><font color="Green">E148</font></i></a>, <a class="ref" href="graphsp.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:100_4_2_1_2_1_1_1_1_2"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:100_4_2_1_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E19:100"><i><font color="Green">E99</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4_2_1_2_1_1_1"><i><font color="Green">E144</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:100_4_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E149</font></i>: 
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:100"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E15:100"><i><font color="Green">E95</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4_2_1_2_1_1_1"><i><font color="Green">E144</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:100_4_2_1_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100_4_2_1_2_1_1_1"><i><font color="Green">E142</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4_2_1_2_1_1_1"><i><font color="Green">E144</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4_2_1_2_1_1_1_1_2"><i><font color="Green">E149</font></i></a></i>;<br/><a NAME="E5:100_4_2_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graphsp.html#E4:100_4_2_1_2_1_1_1"><i><font color="Green">E144</font></i></a>, <a class="txt" href="graphsp.html#E5:100_4_2_1_2_1_1_1"><i><font color="Green">E145</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4_2_1_2_1_1_1_1_2"><i><font color="Green">E149</font></i></a></i>;<br/><a NAME="E6:100_4_2_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E7:100_4_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E150</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E25:100"><i><font color="Green">E102</font></i></a></i>;<br/><a NAME="E8:100_4_2_1_2_1_1_1_1_2"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E27:100"><i><font color="Green">E104</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E9:100_4_2_1_2_1_1_1_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>34</sub></font> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graphsp.html#E4:100_4_2_1_2_1_1_1"><i><font color="Green">E144</font></i></a>, <a class="txt" href="graphsp.html#E82"><i><font color="Green">Lemma72</font></i></a></i>;<br/><a NAME="E10:100_4_2_1_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:100"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E15:100"><i><font color="Green">E95</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E11:100_4_2_1_2_1_1_1_1_2"/>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E4:100_4_2_1"><i><font color="Green">E140</font></i></a>, <a class="txt" href="graphsp.html#E7:100_4_2_1_2_1_1_1"><i><font color="Green">E146</font></i></a>, <a class="txt" href="graphsp.html#E8:100_4_2_1_2_1_1_1"><i><font color="Green">E147</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_1_2_1_1_1_1_2"><i><font color="Green">E148</font></i></a>, <a class="txt" href="graphsp.html#E7:100_4_2_1_2_1_1_1_1_2"><i><font color="Green">E150</font></i></a>, <a class="txt" href="graphsp.html#E8:100_4_2_1_2_1_1_1_1_2"><i><font color="Green">E151</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:100_4_2_1_2_1_1"/>
<font color="Maroon">c<sub>31</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>23</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E12:100"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graphsp.html#E18:100"><i><font color="Green">E98</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4_2_1"><i><font color="Green">E140</font></i></a>, <a class="ref" href="graph_5.html#T68">GRAPH_5:68</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:100_4_2_1_2_1_2"/><i><font color="Green">E141</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><div class="add"><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_1_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_4_2_1_2_1_2_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:100_4_2_1_2_1_2_1_1_1"/><i><font color="Green">E142</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:100_4_2_1_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E143</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graphsp.html#K18" target="_self">newpathcost</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E6:100_4"><i><font color="Green">E120</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_1"><i><font color="Green">E138</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_1_2_1_2"><i><font color="Green">E141</font></i></a>, <a class="ref" href="graphsp.html#D13" target="_self">Def13</a></i>;<br/><a NAME="E3:100_4_2_1_2_1_2_1_1_1"/><i><font color="Green">E144</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E15:100_4"><i><font color="Green">E127</font></i></a>, <a class="txt" href="graphsp.html#E19:100_4"><i><font color="Green">E129</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4_2_1"><i><font color="Green">E140</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>32</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:100_4_2_1_2_1_2_1_1_1"/><i><font color="Green">E145</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E26:100_4"><i><font color="Green">E136</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_1_2_1_2_1_1_1"><i><font color="Green">E142</font></i></a>, <a class="ref" href="graphsp.html#T24" target="_self">Th24</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>33</sub></font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>32</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> as  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="graphsp.html#E5:100_4_2_1_2_1_2_1_1_1"><i><font color="Green">E145</font></i></a>, <a class="ref" href="graphsp.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E7:100_4_2_1_2_1_2_1_1_1"/><i><font color="Green">E146</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>34</sub></font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E9:100_4_2_1_2_1_2_1_1_1"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="graph_5.html#R1">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E30:100"><i><font color="Green">E106</font></i></a>, <a class="txt" href="graphsp.html#E32:100"><i><font color="Green">E107</font></i></a>, <a class="txt" href="graphsp.html#E5:100_4_2_1_2_1_2_1_1_1"><i><font color="Green">E145</font></i></a>, <a class="ref" href="graph_5.html#T37">GRAPH_5:37</a></i>;<br/> <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E13:100"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graphsp.html#E7:100_4_2_1_2_1_2_1_1_1"><i><font color="Green">E146</font></i></a>, <a class="ref" href="graphsp.html#T4" target="_self">Th4</a></i>
<br/>.= 
 <a href="graphsp.html#K13" target="_self">Weight</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E5:100_4_2_1_2_1_2_1_1_1"><i><font color="Green">E145</font></i></a>, <a class="txt" href="graphsp.html#E7:100_4_2_1_2_1_2_1_1_1"><i><font color="Green">E146</font></i></a>, <a class="ref" href="graphsp.html#T26" target="_self">Th26</a></i>
;<br/><a NAME="E11:100_4_2_1_2_1_2_1_1_1"/><b>then </b>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="graphsp.html#K18" target="_self">newpathcost</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E13:100"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graphsp.html#E34:100"><i><font color="Green">E109</font></i></a>, <a class="txt" href="graphsp.html#E35:100"><i><font color="Green">E110</font></i></a>, <a class="txt" href="graphsp.html#E26:100_4"><i><font color="Green">E136</font></i></a>, <a class="txt" href="graphsp.html#E9:100_4_2_1_2_1_2_1_1_1"><i><font color="Green">E147</font></i></a>, <a class="ref" href="graph_5.html#T58">GRAPH_5:58</a></i>;<br/><b>hence </b><a NAME="E12:100_4_2_1_2_1_2_1_1_1"/>
<font color="Maroon">c<sub>31</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>23</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E12:100"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graphsp.html#E17:100"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graphsp.html#E23:100"><i><font color="Green">E101</font></i></a>, <a class="txt" href="graphsp.html#E32:100"><i><font color="Green">E107</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4_2_1"><i><font color="Green">E140</font></i></a>, <a class="txt" href="graphsp.html#E2:100_4_2_1_2_1_2_1_1_1"><i><font color="Green">E143</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4_2_1_2_1_2_1_1_1"><i><font color="Green">E144</font></i></a>, <a class="txt" href="graphsp.html#E5:100_4_2_1_2_1_2_1_1_1"><i><font color="Green">E145</font></i></a>, <a class="txt" href="graphsp.html#E9:100_4_2_1_2_1_2_1_1_1"><i><font color="Green">E147</font></i></a>, <a class="ref" href="graph_5.html#T69">GRAPH_5:69</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:100_4_2_1_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:100_4_2_1_2_1_2_1_1_2"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>6</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E26:100_4"><i><font color="Green">E136</font></i></a>, <a class="ref" href="graphsp.html#T24" target="_self">Th24</a></i>;<br/><b>hence </b><a NAME="E3:100_4_2_1_2_1_2_1_1_2"/>
<font color="Maroon">c<sub>31</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>23</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E12:100"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graphsp.html#E17:100"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graphsp.html#E23:100"><i><font color="Green">E101</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4_2_1"><i><font color="Green">E140</font></i></a>, <a class="ref" href="graphsp.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>thus </b><a NAME="E9:100_4_2_1"/>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E9:100"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graphsp.html#E6:100_4"><i><font color="Green">E120</font></i></a>, <a class="txt" href="graphsp.html#E9:100_4"><i><font color="Green">E122</font></i></a>, <a class="txt" href="graphsp.html#E11:100_4"><i><font color="Green">E123</font></i></a>, <a class="txt" href="graphsp.html#E12:100_4"><i><font color="Green">E124</font></i></a>, <a class="txt" href="graphsp.html#E15:100_4"><i><font color="Green">E127</font></i></a>, <a class="txt" href="graphsp.html#E19:100_4"><i><font color="Green">E129</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_1"><i><font color="Green">E138</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4_2_1"><i><font color="Green">E140</font></i></a>, <a class="ref" href="graphsp.html#D14" target="_self">Def14</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:100_4_2_1_3"/><i><font color="Green">E141</font></i>: 
not <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:100_4_2_1_3"/><b>then </b><i><font color="Green">E142</font></i>: 
not <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E17:100"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_4_2_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>32</sub></font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>assume </b><a NAME="E1:100_4_2_1_3_1"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_4_2_1_3_1_1"/>
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="graphsp.html#E17:100"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_1_3_1"><i><font color="Green">E143</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:100_4_2_1_3_1_1_1"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:100_4_2_1_3_1_1_1"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>take </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>19</sub></font>;<br/><b>thus </b><a NAME="E3:100_4_2_1_3_1_1_1"/>
<font color="Maroon">c<sub>33</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>32</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E17:100"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graphsp.html#E23:100"><i><font color="Green">E101</font></i></a>, <a class="txt" href="graphsp.html#E2:100_4_2_1_3_1_1_1"><i><font color="Green">E144</font></i></a>, <a class="ref" href="graphsp.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E4:100_4_2_1_3_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E3:100_4"><i><font color="Green">E117</font></i></a>, <a class="txt" href="graphsp.html#E21:100_4"><i><font color="Green">E131</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graphsp.html#E2:100_4_2_1_3"><i><font color="Green">E142</font></i></a></i>;<br/><a NAME="E5:100_4_2_1_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100_4"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4"><i><font color="Green">E117</font></i></a>, <a class="txt" href="graphsp.html#E21:100_4"><i><font color="Green">E131</font></i></a>, <a class="txt" href="graphsp.html#E2:100_4_2_1"><i><font color="Green">E139</font></i></a></i>;<br/><a NAME="E6:100_4_2_1_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E7:100_4_2_1_3_1_1_1"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E25:100"><i><font color="Green">E102</font></i></a></i>;<br/><a NAME="E8:100_4_2_1_3_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E23:100"><i><font color="Green">E101</font></i></a>, <a class="txt" href="graphsp.html#E27:100"><i><font color="Green">E104</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E9:100_4_2_1_3_1_1_1"/>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E14:100_4"><i><font color="Green">E126</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4_2_1"><i><font color="Green">E140</font></i></a>, <a class="txt" href="graphsp.html#E7:100_4_2_1_3_1_1_1"><i><font color="Green">E145</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:100_4_2_1_3_1_1_2"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>33</sub></font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E3:100_4_2_1_3_1_1_2"/><i><font color="Green">E145</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>32</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E17:100"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4_2_1"><i><font color="Green">E140</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_1_3"><i><font color="Green">E141</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_1_3_1"><i><font color="Green">E143</font></i></a>, <a class="ref" href="graph_5.html#D19">GRAPH_5:def 19</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><i><font color="Green">E146</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="100_4_2_1_3_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>34</sub></font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>35</sub></font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>assume </b><a NAME="E1:100_4_2_1_3_1_1_2_1"/><i><font color="Green">E147</font></i>: 
( not <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>35</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 ;<br/><a NAME="E2:100_4_2_1_3_1_1_2_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>35</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E4:100_4_2_1_3_1_1_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E10:100"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graphsp.html#E2:100_4_2_1_3_1_1_2_1"><i><font color="Green">E148</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:100_4_2_1_3_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E3:100"><i><font color="Green">E85</font></i></a>, <a class="txt" href="graphsp.html#E12:100"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_1_3_1_1_2_1"><i><font color="Green">E147</font></i></a>, <a class="txt" href="graphsp.html#E92"><i><font color="Green">Lemma81</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>37</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>38</sub></font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E7:100_4_2_1_3_1_1_2_1"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">c<sub>37</sub></font> <a href="graphsp.html#R4" target="_self">is_simple_vertex_seq_at</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>37</sub></font> holds <br/><font color="Maroon">c<sub>37</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>38</sub></font> <a href="graphsp.html#R5" target="_self">is_oriented_edge_seq_of</a> <font color="Maroon">c<sub>37</sub></font> &amp; <font color="Maroon">c<sub>38</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>35</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>38</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> &amp; ( not <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>38</sub></font> <a href="graph_5.html#R9">islongestInShortestpath</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> ) )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E2:100"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_1_3_1_1_2_1"><i><font color="Green">E147</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>39</sub></font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E9:100_4_2_1_3_1_1_2_1"/><i><font color="Green">E150</font></i>: 
( <font color="Maroon">c<sub>39</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>32</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>39</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>38</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100_4_2_1_3_1"><i><font color="Green">E143</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_1_3_1_1_2"><i><font color="Green">E144</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_1_3_1_1_2_1"><i><font color="Green">E147</font></i></a>, <a class="txt" href="graphsp.html#E7:100_4_2_1_3_1_1_2_1"><i><font color="Green">E149</font></i></a>, <a class="ref" href="graph_5.html#D19">GRAPH_5:def 19</a></i>;<br/><a NAME="E10:100_4_2_1_3_1_1_2_1"/>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>39</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:100_4_2_1_3_1_1_2"><i><font color="Green">E145</font></i></a>, <a class="txt" href="graphsp.html#E9:100_4_2_1_3_1_1_2_1"><i><font color="Green">E150</font></i></a>, <a class="ref" href="graphsp.html#T9" target="_self">Th9</a></i>;<br/><b>hence </b><a NAME="E11:100_4_2_1_3_1_1_2_1"/>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100_4_2_1_3_1_1_2_1"><i><font color="Green">E147</font></i></a>, <a class="txt" href="graphsp.html#E7:100_4_2_1_3_1_1_2_1"><i><font color="Green">E149</font></i></a>, <a class="txt" href="graphsp.html#E9:100_4_2_1_3_1_1_2_1"><i><font color="Green">E150</font></i></a>, <a class="ref" href="graphsp.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div><b>take </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>33</sub></font>;<br/><b>thus </b><a NAME="E5:100_4_2_1_3_1_1_2"/>
<font color="Maroon">c<sub>34</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>32</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E12:100"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graphsp.html#E18:100"><i><font color="Green">E98</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_1_3_1"><i><font color="Green">E143</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4_2_1_3_1_1_2"><i><font color="Green">E145</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4_2_1_3_1_1_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="graph_5.html#T68">GRAPH_5:68</a></i>;<br/><b>thus </b><a NAME="E6:100_4_2_1_3_1_1_2"/>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:100_4_2_1_3_1_1_2"><i><font color="Green">E145</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:100_4_2_1_3"/>
<font color="Maroon">c<sub>31</sub></font> <a href="graph_5.html#R9">islongestInShortestpath</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="graph_5.html#D19">GRAPH_5:def 19</a></i>;<br/></div>
<b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_2"><b>suppose </b></a><a NAME="E1:100_4_2_2"/><i><font color="Green">E138</font></i>: 
<span class="p1">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:100_4_2_2"/><b>then </b><i><font color="Green">E139</font></i>: 
<span class="p1">(<span class="default"><a href="graphsp.html#K19" target="_self">Relax</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E7:100"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graphsp.html#E34:100"><i><font color="Green">E109</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4"><i><font color="Green">E118</font></i></a>, <a class="txt" href="graphsp.html#E5:100_4"><i><font color="Green">E119</font></i></a>, <a class="txt" href="graphsp.html#E8:100_4"><i><font color="Green">E121</font></i></a>, <a class="ref" href="graphsp.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E3:100_4_2_2"/><i><font color="Green">E140</font></i>: 
( ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="graphsp.html#K18" target="_self">newpathcost</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>24</sub></font> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 )
 <b>by </b><i><a class="txt" href="graphsp.html#E6:100_4"><i><font color="Green">E120</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2"><i><font color="Green">E138</font></i></a>, <a class="ref" href="graphsp.html#D13" target="_self">Def13</a></i>;<br/><a NAME="E4:100_4_2_2"/><i><font color="Green">E141</font></i>: 
 <a href="graphsp.html#K18" target="_self">newpathcost</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K13" target="_self">Weight</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E34:100"><i><font color="Green">E109</font></i></a>, <a class="txt" href="graphsp.html#E26:100_4"><i><font color="Green">E136</font></i></a></i>;<br/><div><i><font color="Green">E142</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="100_4_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:100_4_2_2_1"/><i><font color="Green">E143</font></i>: 
 <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font>
 ;<br/><a NAME="E2:100_4_2_2_1"/><b>then </b><i><font color="Green">E144</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E6:100"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graphsp.html#E16:100"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4"><i><font color="Green">E118</font></i></a>, <a class="txt" href="graphsp.html#E5:100_4"><i><font color="Green">E119</font></i></a>, <a class="txt" href="graphsp.html#E8:100_4"><i><font color="Green">E121</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E3:100_4_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K13" target="_self">Weight</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
 <b>by </b><i><a class="txt" href="graphsp.html#E26:100_4"><i><font color="Green">E136</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4_2_2"><i><font color="Green">E140</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E4:100_4_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E6:100_4"><i><font color="Green">E120</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2"><i><font color="Green">E138</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4_2_2"><i><font color="Green">E141</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2_1"><i><font color="Green">E143</font></i></a>, <a class="txt" href="graphsp.html#E2:100_4_2_2_1"><i><font color="Green">E144</font></i></a>, <a class="ref" href="graphsp.html#D13" target="_self">Def13</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E143</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="100_4_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:100_4_2_2_2"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:100_4_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>28</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:100_4_2_2_2"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 ;<br/><b>thus </b><a NAME="E5:100_4_2_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E19:100"><i><font color="Green">E99</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4_2_2"><i><font color="Green">E140</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4_2_2_2"><i><font color="Green">E144</font></i></a>, <a class="ref" href="graphsp.html#T33" target="_self">Th33</a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>28</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:100_4_2_2"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E26:100_4"><i><font color="Green">E136</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4_2_2"><i><font color="Green">E140</font></i></a>, <a class="ref" href="graphsp.html#T24" target="_self">Th24</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>28</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> as  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="graphsp.html#E8:100_4_2_2"><i><font color="Green">E144</font></i></a>, <a class="ref" href="graphsp.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E10:100_4_2_2"/><i><font color="Green">E145</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>30</sub></font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E12:100_4_2_2"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>30</sub></font> <a href="graph_5.html#R1">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E30:100"><i><font color="Green">E106</font></i></a>, <a class="txt" href="graphsp.html#E32:100"><i><font color="Green">E107</font></i></a>, <a class="txt" href="graphsp.html#E8:100_4_2_2"><i><font color="Green">E144</font></i></a>, <a class="ref" href="graph_5.html#T37">GRAPH_5:37</a></i>;<br/><b>take </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>24</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>;<br/><b>take </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>30</sub></font>;<br/><b>thus </b><a NAME="E13:100_4_2_2"/>
<font color="Maroon">c<sub>31</sub></font> <a href="graphsp.html#R4" target="_self">is_simple_vertex_seq_at</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E5:100"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E9:100"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graphsp.html#E23:100"><i><font color="Green">E101</font></i></a>, <a class="txt" href="graphsp.html#E43:100"><i><font color="Green">E115</font></i></a>, <a class="txt" href="graphsp.html#E2:100_4_2_2"><i><font color="Green">E139</font></i></a>, <a class="txt" href="graphsp.html#E5:100_4_2_2"><i><font color="Green">E142</font></i></a>, <a class="txt" href="graphsp.html#E6:100_4_2_2"><i><font color="Green">E143</font></i></a>, <a class="txt" href="graphsp.html#E91"><i><font color="Green">Lemma80</font></i></a></i>;<br/><a NAME="E14:100_4_2_2"/><i><font color="Green">E147</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>18</sub></font> <a href="graphsp.html#R3" target="_self">is_vertex_seq_at</a> <font color="Maroon">c<sub>4</sub></font>, <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E23:100"><i><font color="Green">E101</font></i></a>, <a class="ref" href="graphsp.html#D18" target="_self">Def18</a></i>;<br/><b>then </b><i><font color="Green">E148</font></i>: <font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> = 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E6"><i><font color="Green">Lemma6</font></i></a></i>
<br/>.= 
 <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E14:100_4_2_2"><i><font color="Green">E147</font></i></a>, <a class="ref" href="graphsp.html#D17" target="_self">Def17</a></i>
;<br/><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>33</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:100_4_2_2_4"/><i><font color="Green">E149</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>31</sub></font> )
 ;<br/><a NAME="E2:100_4_2_2_4"/><b>then </b>
<font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/><a NAME="E3:100_4_2_2_4"/><b>then </b><i><font color="Green">E150</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_4_2_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_4_2_2_4_1_1"/>
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> or <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:100_4_2_2_4_1_1_1"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:100_4_2_2_4_1_1_1"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E15:100_4_2_2"><i><font color="Green">E148</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:100_4_2_2_4_1_1_2"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><div class="add"><a NAME="E2:100_4_2_2_4_1_1_2"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:100_4_2_2_4"><i><font color="Green">E150</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:100_4_2_2_4_1_1_2"/>
<font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100_4_2_2_4"><i><font color="Green">E149</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4_2_2_4"><i><font color="Green">E150</font></i></a>, <a class="txt" href="graphsp.html#E6"><i><font color="Green">Lemma6</font></i></a></i>;<br/><b>hence </b><a NAME="E4:100_4_2_2_4_1_1_2"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E23:100"><i><font color="Green">E101</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2_4"><i><font color="Green">E149</font></i></a>, <a class="txt" href="graphsp.html#E2:100_4_2_2_4_1_1_2"><i><font color="Green">E151</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:100_4_2_2_4"/>
<font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E17:100"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div>
<b>end;</b></div><a NAME="E17:100_4_2_2"/><i><font color="Green">E149</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E10:100_4_2_2"><i><font color="Green">E145</font></i></a>, <a class="txt" href="graphsp.html#E12:100_4_2_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E18:100_4_2_2"/><i><font color="Green">E150</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E23:100"><i><font color="Green">E101</font></i></a>, <a class="ref" href="graphsp.html#D19" target="_self">Def19</a></i>;<br/><a NAME="E19:100_4_2_2"/><b>then </b><i><font color="Green">E151</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>32</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E17:100_4_2_2"><i><font color="Green">E149</font></i></a>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/><b>set </b><font color="Maroon">c<sub>33</sub></font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>34</sub></font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_4_2_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>35</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:100_4_2_2_5"/><i><font color="Green">E152</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>32</sub></font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_4_2_2_5_1"/>
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>32</sub></font> or <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>32</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_2_5_1_1"><b>suppose </b></a><a NAME="E1:100_4_2_2_5_1_1"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>32</sub></font>
 ;<br/><div class="add"><a NAME="E2:100_4_2_2_5_1_1"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E10:100_4_2_2"><i><font color="Green">E145</font></i></a>, <a class="txt" href="graphsp.html#E12:100_4_2_2"><i><font color="Green">E146</font></i></a>, <a class="txt" href="graphsp.html#E17:100_4_2_2"><i><font color="Green">E149</font></i></a>, <a class="txt" href="graphsp.html#E7"><i><font color="Green">Lemma7</font></i></a></i>;<br/><a NAME="E3:100_4_2_2_5_1_1"/><b>then </b><i><font color="Green">E155</font></i>: 
( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E8:100_4_2_2"><i><font color="Green">E144</font></i></a>, <a class="ref" href="graph_4.html#D1">GRAPH_4:def 1</a></i>;<br/><b>thus </b><a NAME="E4:100_4_2_2_5_1_1"/>
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E8:100_4_2_2"><i><font color="Green">E144</font></i></a>, <a class="txt" href="graphsp.html#E15:100_4_2_2"><i><font color="Green">E148</font></i></a>, <a class="txt" href="graphsp.html#E17:100_4_2_2"><i><font color="Green">E149</font></i></a>, <a class="txt" href="graphsp.html#E18:100_4_2_2"><i><font color="Green">E150</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2_5_1_1"><i><font color="Green">E153</font></i></a>, <a class="txt" href="graphsp.html#E2:100_4_2_2_5_1_1"><i><font color="Green">E154</font></i></a>, <a class="ref" href="graph_4.html#D1">GRAPH_4:def 1</a></i>;<br/><b>thus </b><a NAME="E5:100_4_2_2_5_1_1"/>
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100_4"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graphsp.html#E17:100_4_2_2"><i><font color="Green">E149</font></i></a>, <a class="txt" href="graphsp.html#E18:100_4_2_2"><i><font color="Green">E150</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2_5_1_1"><i><font color="Green">E153</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4_2_2_5_1_1"><i><font color="Green">E155</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_2_5_1_2"><b>suppose </b></a><a NAME="E1:100_4_2_2_5_1_2"/>
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>32</sub></font>
 ;<br/><div class="add"><a NAME="E2:100_4_2_2_5_1_2"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100_4_2_2_5"><i><font color="Green">E152</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:100_4_2_2_5_1_2"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E17:100_4_2_2"><i><font color="Green">E149</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:100_4_2_2_5_1_2"/><b>then </b><i><font color="Green">E155</font></i>: 
( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="graphsp.html#E23:100"><i><font color="Green">E101</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2_5"><i><font color="Green">E152</font></i></a>, <a class="ref" href="graphsp.html#D19" target="_self">Def19</a></i>;<br/><a NAME="E5:100_4_2_2_5_1_2"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E12:100_4_2_2"><i><font color="Green">E146</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2_5"><i><font color="Green">E152</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4_2_2_5_1_2"><i><font color="Green">E154</font></i></a>, <a class="txt" href="graphsp.html#E6"><i><font color="Green">Lemma6</font></i></a></i>;<br/><a NAME="E6:100_4_2_2_5_1_2"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E17:100_4_2_2"><i><font color="Green">E149</font></i></a>, <a class="txt" href="graphsp.html#E18:100_4_2_2"><i><font color="Green">E150</font></i></a>, <a class="txt" href="graphsp.html#E2:100_4_2_2_5_1_2"><i><font color="Green">E153</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>thus </b><a NAME="E7:100_4_2_2_5_1_2"/>
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E17:100_4_2_2"><i><font color="Green">E149</font></i></a>, <a class="txt" href="graphsp.html#E18:100_4_2_2"><i><font color="Green">E150</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2_5"><i><font color="Green">E152</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4_2_2_5_1_2"><i><font color="Green">E155</font></i></a>, <a class="txt" href="graphsp.html#E5:100_4_2_2_5_1_2"><i><font color="Green">E156</font></i></a>, <a class="txt" href="graphsp.html#E6"><i><font color="Green">Lemma6</font></i></a></i>;<br/><a NAME="E8:100_4_2_2_5_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E9:100_4_2_2_5_1_2"/>
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E4:100_4_2_2_5_1_2"><i><font color="Green">E155</font></i></a>, <a class="txt" href="graphsp.html#E5:100_4_2_2_5_1_2"><i><font color="Green">E156</font></i></a>, <a class="txt" href="graphsp.html#E6:100_4_2_2_5_1_2"><i><font color="Green">E157</font></i></a>, <a class="txt" href="graphsp.html#E6"><i><font color="Green">Lemma6</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E21:100_4_2_2"/>
<font color="Maroon">c<sub>32</sub></font> <a href="graphsp.html#R5" target="_self">is_oriented_edge_seq_of</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E19:100_4_2_2"><i><font color="Green">E151</font></i></a>, <a class="ref" href="graphsp.html#D19" target="_self">Def19</a></i>;<br/><a NAME="E22:100_4_2_2"/><i><font color="Green">E152</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E13:100"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graphsp.html#E12:100_4_2_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="graph_5.html#T58">GRAPH_5:58</a></i>;<br/><i><font color="Green">E153</font></i>:  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E13:100"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graphsp.html#E10:100_4_2_2"><i><font color="Green">E145</font></i></a>, <a class="ref" href="graphsp.html#T4" target="_self">Th4</a></i>
<br/>.= 
 <a href="graphsp.html#K13" target="_self">Weight</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E8:100_4_2_2"><i><font color="Green">E144</font></i></a>, <a class="txt" href="graphsp.html#E10:100_4_2_2"><i><font color="Green">E145</font></i></a>, <a class="ref" href="graphsp.html#T26" target="_self">Th26</a></i>
;<br/><a NAME="E24:100_4_2_2"/><b>then </b><i><font color="Green">E154</font></i>: 
 <a href="graphsp.html#K18" target="_self">newpathcost</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E34:100"><i><font color="Green">E109</font></i></a>, <a class="txt" href="graphsp.html#E35:100"><i><font color="Green">E110</font></i></a>, <a class="txt" href="graphsp.html#E26:100_4"><i><font color="Green">E136</font></i></a>, <a class="txt" href="graphsp.html#E22:100_4_2_2"><i><font color="Green">E152</font></i></a></i>;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_2_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_4_2_2_7_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 or <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_2_7_1_1"><b>suppose </b></a><a NAME="E1:100_4_2_2_7_1_1"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_4_2_2_7_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>35</sub></font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>assume </b><a NAME="E1:100_4_2_2_7_1_1_1"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>35</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/> <a href="real_1.html#K1">-</a> 1 = 
<font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>36</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E3:100"><i><font color="Green">E85</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4"><i><font color="Green">E117</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2_7_1_1"><i><font color="Green">E155</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2_7_1_1_1"><i><font color="Green">E156</font></i></a></i>
<br/>.= 
 <a href="graphsp.html#K13" target="_self">Weight</a> <font color="Maroon">c<sub>35</sub></font>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4"><i><font color="Green">E116</font></i></a>, <a class="ref" href="graphsp.html#D20" target="_self">Def20</a></i>
;<br/><b>hence </b><a NAME="E4:100_4_2_2_7_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>6</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">c<sub>35</sub></font>,<font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="ref" href="graphsp.html#T24" target="_self">Th24</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:100_4_2_2_7_1_1"/>
<font color="Maroon">c<sub>32</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>23</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E17:100"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graphsp.html#E23:100"><i><font color="Green">E101</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graphsp.html#E8:100_4_2_2"><i><font color="Green">E144</font></i></a>, <a class="txt" href="graphsp.html#E12:100_4_2_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="graphsp.html#T15" target="_self">Th15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_2_7_1_2"><b>suppose </b></a><a NAME="E1:100_4_2_2_7_1_2"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>35</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>36</sub></font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E3:100_4_2_2_7_1_2"/><i><font color="Green">E156</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="graphsp.html#R4" target="_self">is_simple_vertex_seq_at</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>35</sub></font> holds <br/><font color="Maroon">c<sub>35</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>36</sub></font> <a href="graphsp.html#R5" target="_self">is_oriented_edge_seq_of</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>23</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> &amp; ( not <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>36</sub></font> <a href="graph_5.html#R9">islongestInShortestpath</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> ) )
 <b>by </b><i><a class="txt" href="graphsp.html#E2:100"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4"><i><font color="Green">E116</font></i></a></i>;<br/><b>thus </b><a NAME="E4:100_4_2_2_7_1_2"/>
<font color="Maroon">c<sub>32</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>23</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E6:100"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graphsp.html#E12:100"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graphsp.html#E16:100"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graphsp.html#E17:100"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graphsp.html#E23:100"><i><font color="Green">E101</font></i></a>, <a class="txt" href="graphsp.html#E32:100"><i><font color="Green">E107</font></i></a>, <a class="txt" href="graphsp.html#E35:100"><i><font color="Green">E110</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4"><i><font color="Green">E118</font></i></a>, <a class="txt" href="graphsp.html#E5:100_4"><i><font color="Green">E119</font></i></a>, <a class="txt" href="graphsp.html#E8:100_4"><i><font color="Green">E121</font></i></a>, <a class="txt" href="graphsp.html#E27:100_4"><i><font color="Green">E137</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4_2_2"><i><font color="Green">E140</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4_2_2"><i><font color="Green">E141</font></i></a>, <a class="txt" href="graphsp.html#E8:100_4_2_2"><i><font color="Green">E144</font></i></a>, <a class="txt" href="graphsp.html#E12:100_4_2_2"><i><font color="Green">E146</font></i></a>, <a class="txt" href="graphsp.html#E22:100_4_2_2"><i><font color="Green">E152</font></i></a>, <a class="txt" href="graphsp.html#E23:100_4_2_2"><i><font color="Green">E153</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2_7_1_2"><i><font color="Green">E155</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4_2_2_7_1_2"><i><font color="Green">E156</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self">Th19</a>, <a class="ref" href="graph_5.html#T69">GRAPH_5:69</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>thus </b><a NAME="E26:100_4_2_2"/>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E9:100"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graphsp.html#E6:100_4"><i><font color="Green">E120</font></i></a>, <a class="txt" href="graphsp.html#E9:100_4"><i><font color="Green">E122</font></i></a>, <a class="txt" href="graphsp.html#E11:100_4"><i><font color="Green">E123</font></i></a>, <a class="txt" href="graphsp.html#E12:100_4"><i><font color="Green">E124</font></i></a>, <a class="txt" href="graphsp.html#E15:100_4"><i><font color="Green">E127</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2"><i><font color="Green">E138</font></i></a>, <a class="txt" href="graphsp.html#E24:100_4_2_2"><i><font color="Green">E154</font></i></a>, <a class="ref" href="graphsp.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E27:100_4_2_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E12:100"><i><font color="Green">E92</font></i></a>, <a class="ref" href="graph_5.html#T54">GRAPH_5:54</a></i>;<br/><a NAME="E28:100_4_2_2"/><b>then </b><i><font color="Green">E155</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E22:100_4_2_2"><i><font color="Green">E152</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:100_4_2_2_8"/>
not <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_4_2_2_8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>35</sub></font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>assume </b><a NAME="E1:100_4_2_2_8_1"/><i><font color="Green">E156</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_2_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_4_2_2_8_1_1"/>
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="graphsp.html#E17:100"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2_8_1"><i><font color="Green">E156</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_2_8_1_1_1"><b>suppose </b></a><a NAME="E1:100_4_2_2_8_1_1_1"/>
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:100_4_2_2_8_1_1_1"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a>  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>take </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>19</sub></font>;<br/><b>thus </b><a NAME="E3:100_4_2_2_8_1_1_1"/>
<font color="Maroon">c<sub>36</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>35</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E17:100"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graphsp.html#E23:100"><i><font color="Green">E101</font></i></a>, <a class="txt" href="graphsp.html#E2:100_4_2_2_8_1_1_1"><i><font color="Green">E157</font></i></a>, <a class="ref" href="graphsp.html#T8" target="_self">Th8</a></i>;<br/><b>thus </b><a NAME="E4:100_4_2_2_8_1_1_1"/>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E28:100_4_2_2"><i><font color="Green">E155</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_4_2_2_8_1_1_2"><b>suppose </b></a><a NAME="E1:100_4_2_2_8_1_1_2"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>36</sub></font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E3:100_4_2_2_8_1_1_2"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">c<sub>36</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>35</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E23:100"><i><font color="Green">E101</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2_8_1"><i><font color="Green">E156</font></i></a>, <a class="ref" href="graphsp.html#T40" target="_self">Th40</a>, <a class="ref" href="graph_5.html#D19">GRAPH_5:def 19</a></i>;<br/><div><i><font color="Green">E159</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="100_4_2_2_8_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>37</sub></font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>38</sub></font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>assume </b><a NAME="E1:100_4_2_2_8_1_1_2_1"/><i><font color="Green">E160</font></i>: 
( not <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>38</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 ;<br/><a NAME="E2:100_4_2_2_8_1_1_2_1"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>39</sub></font> = <font color="Maroon">c<sub>38</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E4:100_4_2_2_8_1_1_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>39</sub></font> &amp; <font color="Maroon">c<sub>39</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E10:100"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graphsp.html#E2:100_4_2_2_8_1_1_2_1"><i><font color="Green">E161</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:100_4_2_2_8_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>39</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E3:100"><i><font color="Green">E85</font></i></a>, <a class="txt" href="graphsp.html#E12:100"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2_8_1_1_2_1"><i><font color="Green">E160</font></i></a>, <a class="txt" href="graphsp.html#E92"><i><font color="Green">Lemma81</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>40</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>41</sub></font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E7:100_4_2_2_8_1_1_2_1"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">c<sub>40</sub></font> <a href="graphsp.html#R4" target="_self">is_simple_vertex_seq_at</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>39</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>40</sub></font> holds <br/><font color="Maroon">c<sub>40</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>41</sub></font> <a href="graphsp.html#R5" target="_self">is_oriented_edge_seq_of</a> <font color="Maroon">c<sub>40</sub></font> &amp; <font color="Maroon">c<sub>41</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>38</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>39</sub></font></span>)</span> &amp; ( not <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>41</sub></font> <a href="graph_5.html#R9">islongestInShortestpath</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> ) )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E2:100"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2_8_1_1_2_1"><i><font color="Green">E160</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>42</sub></font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E9:100_4_2_2_8_1_1_2_1"/><i><font color="Green">E163</font></i>: 
( <font color="Maroon">c<sub>42</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>35</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>42</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100_4_2_2_8_1"><i><font color="Green">E156</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2_8_1_1_2"><i><font color="Green">E157</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2_8_1_1_2_1"><i><font color="Green">E160</font></i></a>, <a class="txt" href="graphsp.html#E7:100_4_2_2_8_1_1_2_1"><i><font color="Green">E162</font></i></a>, <a class="ref" href="graph_5.html#D19">GRAPH_5:def 19</a></i>;<br/><a NAME="E10:100_4_2_2_8_1_1_2_1"/>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>42</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:100_4_2_2_8_1_1_2"><i><font color="Green">E158</font></i></a>, <a class="txt" href="graphsp.html#E9:100_4_2_2_8_1_1_2_1"><i><font color="Green">E163</font></i></a>, <a class="ref" href="graphsp.html#T9" target="_self">Th9</a></i>;<br/><b>hence </b><a NAME="E11:100_4_2_2_8_1_1_2_1"/>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100_4_2_2_8_1_1_2_1"><i><font color="Green">E160</font></i></a>, <a class="txt" href="graphsp.html#E7:100_4_2_2_8_1_1_2_1"><i><font color="Green">E162</font></i></a>, <a class="txt" href="graphsp.html#E9:100_4_2_2_8_1_1_2_1"><i><font color="Green">E163</font></i></a>, <a class="ref" href="graphsp.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div><b>take </b><font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>36</sub></font>;<br/><b>thus </b><a NAME="E5:100_4_2_2_8_1_1_2"/>
<font color="Maroon">c<sub>37</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>35</sub></font>, <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E12:100"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graphsp.html#E18:100"><i><font color="Green">E98</font></i></a>, <a class="txt" href="graphsp.html#E1:100_4_2_2_8_1"><i><font color="Green">E156</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4_2_2_8_1_1_2"><i><font color="Green">E158</font></i></a>, <a class="txt" href="graphsp.html#E4:100_4_2_2_8_1_1_2"><i><font color="Green">E159</font></i></a>, <a class="ref" href="graph_5.html#T68">GRAPH_5:68</a></i>;<br/><b>thus </b><a NAME="E6:100_4_2_2_8_1_1_2"/>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E28:100_4_2_2"><i><font color="Green">E155</font></i></a>, <a class="txt" href="graphsp.html#E3:100_4_2_2_8_1_1_2"><i><font color="Green">E158</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:100_4_2_2_8"/>
<font color="Maroon">c<sub>32</sub></font> <a href="graph_5.html#R9">islongestInShortestpath</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="graph_5.html#D19">GRAPH_5:def 19</a></i>;<br/></div>
<b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>24</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:100_5"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>set </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font>;<br/><a NAME="E2:100_5"/><i><font color="Green">E117</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100_5"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graphsp.html#E83"><i><font color="Green">Lemma73</font></i></a></i>;<br/><a NAME="E3:100_5"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:100"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E15:100"><i><font color="Green">E95</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:100_5"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100_5"><i><font color="Green">E116</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E5:100_5"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><i><font color="Green">E120</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="100_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:100_5_1"/>
<span class="p1">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:100_5_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E7:100"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graphsp.html#E9:100"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graphsp.html#E34:100"><i><font color="Green">E109</font></i></a>, <a class="txt" href="graphsp.html#E2:100_5"><i><font color="Green">E117</font></i></a>, <a class="txt" href="graphsp.html#E3:100_5"><i><font color="Green">E118</font></i></a>, <a class="txt" href="graphsp.html#E5:100_5"><i><font color="Green">E119</font></i></a>, <a class="ref" href="graphsp.html#D14" target="_self">Def14</a></i>;<br/><b>hence </b><a NAME="E3:100_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100_5"><i><font color="Green">E116</font></i></a>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/></div><b>end;</b></div><a NAME="E7:100_5"/><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E6:100"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graphsp.html#E16:100"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graphsp.html#E2:100_5"><i><font color="Green">E117</font></i></a>, <a class="txt" href="graphsp.html#E3:100_5"><i><font color="Green">E118</font></i></a>, <a class="txt" href="graphsp.html#E5:100_5"><i><font color="Green">E119</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E8:100_5"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E7:100"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graphsp.html#E9:100"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graphsp.html#E1:100_5"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graphsp.html#E2:100_5"><i><font color="Green">E117</font></i></a>, <a class="txt" href="graphsp.html#E3:100_5"><i><font color="Green">E118</font></i></a>, <a class="txt" href="graphsp.html#E5:100_5"><i><font color="Green">E119</font></i></a>, <a class="txt" href="graphsp.html#E6:100_5"><i><font color="Green">E120</font></i></a>, <a class="ref" href="graphsp.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E9:100_5"/><b>then </b><i><font color="Green">E123</font></i>: 
not <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:100"><i><font color="Green">E85</font></i></a></i>;<br/><a NAME="E10:100_5"/>
<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E19:100"><i><font color="Green">E99</font></i></a>, <a class="txt" href="graphsp.html#E1:100_5"><i><font color="Green">E116</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:100_5"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:100"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E15:100"><i><font color="Green">E95</font></i></a>, <a class="txt" href="graphsp.html#E1:100_5"><i><font color="Green">E116</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:100_5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100_5"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graphsp.html#E11:100_5"><i><font color="Green">E124</font></i></a>, <a class="txt" href="graphsp.html#E9:100_5"><i><font color="Green">E123</font></i></a></i>;<br/><a NAME="E13:100_5"/><b>then </b><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E6:100"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graphsp.html#E16:100"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graphsp.html#E19:100"><i><font color="Green">E99</font></i></a>, <a class="txt" href="graphsp.html#E1:100_5"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graphsp.html#E8:100_5"><i><font color="Green">E122</font></i></a>, <a class="txt" href="graphsp.html#E11:100_5"><i><font color="Green">E124</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E14:100_5"/>
not <span class="p1">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E6:100_5"><i><font color="Green">E120</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E15:100_5"/><b>then </b><i><font color="Green">E126</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="graphsp.html#E7:100_5"><i><font color="Green">E121</font></i></a>, <a class="txt" href="graphsp.html#E8:100_5"><i><font color="Green">E122</font></i></a>, <a class="txt" href="graphsp.html#E13:100_5"><i><font color="Green">E125</font></i></a>, <a class="ref" href="graphsp.html#D13" target="_self">Def13</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>24</sub></font> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="graphsp.html#E10:100"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graphsp.html#E1:100_5"><i><font color="Green">E116</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>set </b><font color="Maroon">c<sub>27</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font>;<br/><a NAME="E17:100_5"/><i><font color="Green">E127</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> &amp;  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graphsp.html#E16:100"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graphsp.html#E1:100_5"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graphsp.html#E84"><i><font color="Green">Lemma74</font></i></a></i>;<br/><a NAME="E18:100_5"/><b>then </b><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:100"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E15:100"><i><font color="Green">E95</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E19:100_5"/><i><font color="Green">E129</font></i>: 
( <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="graphsp.html#E16:100"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graphsp.html#E1:100_5"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graphsp.html#E85"><i><font color="Green">Lemma75</font></i></a></i>;<br/><a NAME="E20:100_5"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p5">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="graphsp.html#K13" target="_self">Weight</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="ref" href="graphsp.html#D20" target="_self">Def20</a></i>;<br/><i><font color="Green">E131</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p5">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E7:100"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graphsp.html#E34:100"><i><font color="Green">E109</font></i></a>, <a class="txt" href="graphsp.html#E18:100_5"><i><font color="Green">E128</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p5">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E6:100"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graphsp.html#E17:100_5"><i><font color="Green">E127</font></i></a>, <a class="txt" href="graphsp.html#E18:100_5"><i><font color="Green">E128</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self">Th19</a></i>
<br/>.= 
 <a href="graphsp.html#K13" target="_self">Weight</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E15:100"><i><font color="Green">E95</font></i></a>, <a class="txt" href="graphsp.html#E28:100"><i><font color="Green">E105</font></i></a>, <a class="txt" href="graphsp.html#E18:100_5"><i><font color="Green">E128</font></i></a>, <a class="txt" href="graphsp.html#E19:100_5"><i><font color="Green">E129</font></i></a>, <a class="txt" href="graphsp.html#E20:100_5"><i><font color="Green">E130</font></i></a>, <a class="ref" href="graphsp.html#D16" target="_self">Def16</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_5_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_5_3"/>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="graphsp.html#E17:100"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graphsp.html#E1:100_5"><i><font color="Green">E116</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_5_3_1"><b>suppose </b></a><a NAME="E1:100_5_3_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:100_5_3_1"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:100_5_3_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>6</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>26</sub></font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E15:100_5"><i><font color="Green">E126</font></i></a>, <a class="txt" href="graphsp.html#E21:100_5"><i><font color="Green">E131</font></i></a>, <a class="ref" href="graphsp.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E4:100_5_3_1"/><b>then </b>
 <a href="graphsp.html#K13" target="_self">Weight</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="graphsp.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E5:100_5_3_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E2:100_5_3_1"><i><font color="Green">E132</font></i></a>, <a class="ref" href="graphsp.html#D20" target="_self">Def20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_5_3_2"><b>suppose </b></a><a NAME="E1:100_5_3_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:100_5_3_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E3:100"><i><font color="Green">E85</font></i></a>, <a class="txt" href="graphsp.html#E1:100_5"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graphsp.html#E8:100_5"><i><font color="Green">E122</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>23</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E46:100"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_6"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_6"/>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i><a class="txt" href="graphsp.html#E17:100"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graphsp.html#E46:100"><i><font color="Green">E116</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_6_1"><b>suppose </b></a><a NAME="E1:100_6_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:100_6_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:100"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E5:100"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E89"><i><font color="Green">Lemma78</font></i></a></i>;<br/><b>hence </b><a NAME="E3:100_6_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E3:100"><i><font color="Green">E85</font></i></a>, <a class="txt" href="graphsp.html#E1:100_6_1"><i><font color="Green">E117</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_6_2"><b>suppose </b></a><a NAME="E1:100_6_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:100_6_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E16:100"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graphsp.html#E43:100"><i><font color="Green">E115</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
