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

<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> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><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="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>8</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>9</sub></font>, <font color="Maroon">c<sub>10</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="E1:99"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c<sub>4</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>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="graph_5.html#R5">is_weight&gt;=0of</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; ( 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>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="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> <font color="Maroon">c<sub>3</sub></font> holds <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> <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 ) )
 ;<br/>

<b>assume </b><a NAME="E2:99"/><i><font color="Green">E83</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>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/>

<a NAME="E3:99"/>
<font color="Maroon">c<sub>7</sub></font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:99"><i><font color="Green">E82</font></i></a>, <a class="ref" href="graph_5.html#D18">GRAPH_5:def 18</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being  <a href="graph_1.html#V8">oriented</a> <a href="graph_4.html#V1">Simple</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="E5:99"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:99"><i><font color="Green">E82</font></i></a>, <a class="ref" href="graph_5.html#T66">GRAPH_5:66</a></i>;<br/>
<a NAME="E6:99"/>
<font color="Maroon">c<sub>11</sub></font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E5:99"><i><font color="Green">E84</font></i></a>, <a class="ref" href="graph_5.html#D18">GRAPH_5:def 18</a></i>;<br/>
<a NAME="E7:99"/><b>then </b><i><font color="Green">E85</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="graph_5.html#R1">is_orientedpath_of</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1">(<span class="default"><a href="graph_5.html#K2">vertices</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="graph_5.html#D4">GRAPH_5:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E8:99"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &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>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="graphsp.html#E7:99"><i><font color="Green">E85</font></i></a>, <a class="ref" href="graph_5.html#D3">GRAPH_5:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E10:99"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="graph_4.html#R3">is_oriented_vertex_seq_of</a> <font color="Maroon">c<sub>11</sub></font> &amp; ( 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 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="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> ) ) )
 <b>by </b><i><a class="ref" href="graph_4.html#D7">GRAPH_4:def 7</a></i>;<br/>
<a NAME="E11:99"/><i><font color="Green">E88</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</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>11</sub></font></span>)</span> <a href="nat_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 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>11</sub></font> holds <br/><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 
<b>by </b><i><a class="txt" href="graphsp.html#E10:99"><i><font color="Green">E87</font></i></a>, <a class="ref" href="graph_4.html#D5">GRAPH_4:def 5</a></i>;<br/>
<a NAME="E12:99"/><i><font color="Green">E89</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="graphsp.html#E8:99"><i><font color="Green">E86</font></i></a>, <a class="ref" href="graph_5.html#T8">GRAPH_5:8</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</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>11</sub></font></span>)</span>;<br/>
<a NAME="E13:99"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>11</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>11</sub></font></span>)</span> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E10:99"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E12:99"><i><font color="Green">E89</font></i></a>, <a class="ref" href="graph_4.html#D5">GRAPH_4:def 5</a></i>;<br/>
<a NAME="E14:99"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E12:99"><i><font color="Green">E89</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E15:99"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>11</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>11</sub></font></span>)</span> <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>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E16:99"/><i><font color="Green">E92</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E11:99"><i><font color="Green">E88</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E17:99"/><i><font color="Green">E93</font></i>: 
1 <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>11</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="graphsp.html#E12:99"><i><font color="Green">E89</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E18:99"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</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>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E12:99"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graphsp.html#E16:99"><i><font color="Green">E92</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</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>11</sub></font></span>)</span> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<a NAME="E20:99"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E8:99"><i><font color="Green">E86</font></i></a>, <a class="txt" href="graphsp.html#E13:99"><i><font color="Green">E90</font></i></a>, <a class="ref" href="graph_4.html#D1">GRAPH_4:def 1</a></i>;<br/>
<a NAME="E21:99"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</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>13</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E11:99"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graphsp.html#E17:99"><i><font color="Green">E93</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<div><i><font color="Green">E97</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:99_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:99_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E10:99"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E12:99"><i><font color="Green">E89</font></i></a>, <a class="ref" href="graph_4.html#D5">GRAPH_4:def 5</a></i>;<br/><font color="Maroon">c<sub>10</sub></font> = 
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>11</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E7:99"><i><font color="Green">E85</font></i></a>, <a class="ref" href="graph_5.html#D3">GRAPH_5:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="graphsp.html#E2:99_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="graph_4.html#D1">GRAPH_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="graphsp.html#E11:99"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graphsp.html#E17:99"><i><font color="Green">E93</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E10:99"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graphsp.html#E12:99"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graphsp.html#E16:99"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graphsp.html#E21:99"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graphsp.html#E1:99_1"><i><font color="Green">E98</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:99_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E1:99"><i><font color="Green">E82</font></i></a>, <a class="txt" href="graphsp.html#E1:99_1"><i><font color="Green">E98</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E23:99"/><i><font color="Green">E98</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#E1:99"><i><font color="Green">E82</font></i></a>, <a class="ref" href="graphsp.html#D20" target="_self">Def20</a></i>;<br/>
<a NAME="E24:99"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E12:99"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graphsp.html#E16:99"><i><font color="Green">E92</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E25:99"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E23:99"><i><font color="Green">E98</font></i></a>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E27:99"/><i><font color="Green">E99</font></i>: 
<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> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="graphsp.html#K13" target="_self">Weight</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:99"><i><font color="Green">E82</font></i></a>, <a class="ref" href="graphsp.html#D20" target="_self">Def20</a></i>;<br/>
<a NAME="E28:99"/><i><font color="Green">E100</font></i>: 
not <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E22:99"><i><font color="Green">E97</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E29:99"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 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>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E13:99"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graphsp.html#E18:99"><i><font color="Green">E94</font></i></a>, <a class="ref" href="graph_4.html#D1">GRAPH_4:def 1</a></i>;<br/>
<a NAME="E30:99"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2">vertices</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E12:99"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graphsp.html#E15"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<a NAME="E31:99"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="graph_5.html#K2">vertices</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E28:99"><i><font color="Green">E100</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E32:99"/><b>then </b>
<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> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</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#E1:99"><i><font color="Green">E82</font></i></a>, <a class="txt" href="graphsp.html#E2:99"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graphsp.html#E7:99"><i><font color="Green">E85</font></i></a></i>;<br/>
<b>hence </b><a NAME="E33:99"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E13:99"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graphsp.html#E15:99"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graphsp.html#E18:99"><i><font color="Green">E94</font></i></a>, <a class="txt" href="graphsp.html#E20:99"><i><font color="Green">E95</font></i></a>, <a class="txt" href="graphsp.html#E27:99"><i><font color="Green">E99</font></i></a>, <a class="ref" href="graphsp.html#T24" target="_self">Th24</a></i>;<br/>


</div>
