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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">V</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ;<br/><b>let </b><font color="Maroon" title="c6">G</font> be  <a href="graph_1.html#V3" title="GRAPH_1:attr.3">oriented</a> <a href="graph_1.html#V7" title="GRAPH_1:attr.7">finite</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c7">P</font> be  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font>;<br/><b>let </b><font color="Maroon" title="c8">W</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c6">G</font>, <a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a> ;<br/><b>let </b><font color="Maroon" title="c9">v2</font>, <font color="Maroon" title="c10">v1</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c6">G</font>;<br/>



















<b>assume </b><a NAME="E1:97"/><i><font color="Green" title="E81">A1</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="graphsp.html#R6" title="GRAPHSP:pred.6">is_Input_of_Dijkstra_Alg</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">W</font> &amp; <font color="Maroon" title="c8">W</font> <a href="graph_5.html#R5" title="GRAPH_5:pred.5">is_weight&gt;=0of</a> <font color="Maroon" title="c6">G</font> &amp; <font color="Maroon" title="c9">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c10">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">v2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c7">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c9">v2</font>,<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c8">W</font> &amp; ( for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) )
 ;<br/>

<b>assume </b><a NAME="E2:97"/><i><font color="Green" title="E82">A2</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ;<br/>

<a NAME="E3:97"/>
<font color="Maroon" title="c7">P</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c9">v2</font>,<font color="Maroon" title="c3">V</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:97"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">q</font> being  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font><b> such that </b><br/><a NAME="E5:97"/><i><font color="Green" title="E83">A3</font></i>: 
<font color="Maroon" title="c11">q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c9">v2</font>,<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c8">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:97"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="graph_5.html#T66" title="GRAPH_5:th.66">GRAPH_5:66</a></i>;<br/>
<a NAME="E6:97"/>
<font color="Maroon" title="c11">q</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c9">v2</font>,<font color="Maroon" title="c3">V</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E5:97"><i><font color="Green" title="E83">A3</font></i></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>;<br/>
<a NAME="E7:97"/><b>then </b><i><font color="Green" title="E84">A4</font></i>: 
( <font color="Maroon" title="c11">q</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c9">v2</font> &amp; <span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c11">q</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">V</font> )
 
<b>by </b><i><a class="ref" href="graph_5.html#D4" title="GRAPH_5:def.4">GRAPH_5:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">FT</font> = the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c6">G</font>;<br/>
<a NAME="E8:97"/><i><font color="Green" title="E85">A5</font></i>: 
( <font color="Maroon" title="c11">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v2</font> )
 
<b>by </b><i><a class="txt" href="graphsp.html#E7:97"><i><font color="Green" title="E84">A4</font></i></a>, <a class="ref" href="graph_5.html#D3" title="GRAPH_5:def.3">GRAPH_5:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">vs</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c6">G</font><b> such that </b><br/><a NAME="E10:97"/><i><font color="Green" title="E86">A6</font></i>: 
( <font color="Maroon" title="c13">vs</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Maroon" title="c11">q</font> &amp; ( for <font color="Olive" title="b1">n1</font>, <font color="Olive" title="b2">m1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n1</font> &amp; <font color="Olive" title="b1">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">m1</font> &amp; <font color="Olive" title="b2">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">vs</font> &amp; <font color="Maroon" title="c13">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m1</font> holds <br/>( <font color="Olive" title="b1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b2">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">vs</font> ) ) )
 <b>by </b><i><a class="ref" href="graph_4.html#D7" title="GRAPH_4:def.7">GRAPH_4:def 7</a></i>;<br/>
<a NAME="E11:97"/><i><font color="Green" title="E87">A7</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">n1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n1</font> &amp; <font color="Olive" title="b1">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font> holds <br/><font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n1</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c13">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n1</font>,<font color="Maroon" title="c13">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) )
 
<b>by </b><i><a class="txt" href="graphsp.html#E10:97"><i><font color="Green" title="E86">A6</font></i></a>, <a class="ref" href="graph_4.html#D5" title="GRAPH_4:def.5">GRAPH_4:def 5</a></i>;<br/>
<a NAME="E12:97"/><i><font color="Green" title="E88">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="graphsp.html#E8:97"><i><font color="Green" title="E85">A5</font></i></a>, <a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">e</font> = <font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font></span>)</span>;<br/>
<a NAME="E13:97"/><i><font color="Green" title="E89">A9</font></i>: 
<font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font></span>)</span> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c13">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font></span>)</span>,<font color="Maroon" title="c13">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E10:97"><i><font color="Green" title="E86">A6</font></i></a>, <a class="txt" href="graphsp.html#E12:97"><i><font color="Green" title="E88">A8</font></i></a>, <a class="ref" href="graph_4.html#D5" title="GRAPH_4:def.5">GRAPH_4:def 5</a></i>;<br/>
<a NAME="E14:97"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">q</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E12:97"><i><font color="Green" title="E88">A8</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E15:97"/><b>then </b><i><font color="Green" title="E90">A10</font></i>: 
<font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c6">G</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E16:97"/><i><font color="Green" title="E91">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">vs</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E11:97"><i><font color="Green" title="E87">A7</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E17:97"/><i><font color="Green" title="E92">A12</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="graphsp.html#E12:97"><i><font color="Green" title="E88">A8</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E18:97"/><i><font color="Green" title="E93">A13</font></i>: 
<font color="Maroon" title="c13">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E12:97"><i><font color="Green" title="E88">A8</font></i></a>, <a class="txt" href="graphsp.html#E16:97"><i><font color="Green" title="E91">A11</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">v3</font> = <font color="Maroon" title="c13">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font></span>)</span> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c6">G</font> ;<br/>
<a NAME="E20:97"/><i><font color="Green" title="E94">A14</font></i>: 
<font color="Maroon" title="c9">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E8:97"><i><font color="Green" title="E85">A5</font></i></a>, <a class="txt" href="graphsp.html#E13:97"><i><font color="Green" title="E89">A9</font></i></a>, <a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></i>;<br/>
<a NAME="E21:97"/><b>then </b><i><font color="Green" title="E95">A15</font></i>: 
<font color="Maroon" title="c9">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">vs</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E11:97"><i><font color="Green" title="E87">A7</font></i></a>, <a class="txt" href="graphsp.html#E17:97"><i><font color="Green" title="E92">A12</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<div><i><font color="Green" title="E96">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_1"/><i><font color="Green" title="E97">A17</font></i>: 
<font color="Maroon" title="c15">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v2</font>
 ;<br/><a NAME="E2:97_1"/><i><font color="Green" title="E98">A18</font></i>: 
<font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c13">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c13">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E10:97"><i><font color="Green" title="E86">A6</font></i></a>, <a class="txt" href="graphsp.html#E12:97"><i><font color="Green" title="E88">A8</font></i></a>, <a class="ref" href="graph_4.html#D5" title="GRAPH_4:def.5">GRAPH_4:def 5</a></i>;<br/><font color="Maroon" title="c10">v1</font> = 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E7:97"><i><font color="Green" title="E84">A4</font></i></a>, <a class="ref" href="graph_5.html#D3" title="GRAPH_5:def.3">GRAPH_5:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c13">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="txt" href="graphsp.html#E2:97_1"><i><font color="Green" title="E98">A18</font></i></a>, <a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c13">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="graphsp.html#E11:97"><i><font color="Green" title="E87">A7</font></i></a>, <a class="txt" href="graphsp.html#E17:97"><i><font color="Green" title="E92">A12</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<font color="Maroon" title="c15">v3</font>
<b>by </b><i><a class="txt" href="graphsp.html#E10:97"><i><font color="Green" title="E86">A6</font></i></a>, <a class="txt" href="graphsp.html#E12:97"><i><font color="Green" title="E88">A8</font></i></a>, <a class="txt" href="graphsp.html#E16:97"><i><font color="Green" title="E91">A11</font></i></a>, <a class="txt" href="graphsp.html#E21:97"><i><font color="Green" title="E95">A15</font></i></a>, <a class="txt" href="graphsp.html#E1:97_1"><i><font color="Green" title="E97">A17</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:97_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E1:97"><i><font color="Green" title="E81">A1</font></i></a>, <a class="txt" href="graphsp.html#E1:97_1"><i><font color="Green" title="E97">A17</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E23:97"/><i><font color="Green" title="E97">A19</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c6">G</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:97"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="graphsp.html#D20" target="_self" title="GRAPHSP:def.20">Def20</a></i>;<br/>
<a NAME="E24:97"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">vs</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E12:97"><i><font color="Green" title="E88">A8</font></i></a>, <a class="txt" href="graphsp.html#E16:97"><i><font color="Green" title="E91">A11</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E25:97"/><b>then </b>
<font color="Maroon" title="c15">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E23:97"><i><font color="Green" title="E97">A19</font></i></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">m</font> = <font color="Maroon" title="c15">v3</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E27:97"/><i><font color="Green" title="E98">A20</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c16">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> <font color="Maroon" title="c15">v3</font>,<font color="Maroon" title="c9">v2</font>,<font color="Maroon" title="c8">W</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:97"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="graphsp.html#D20" target="_self" title="GRAPHSP:def.20">Def20</a></i>;<br/>
<a NAME="E28:97"/><i><font color="Green" title="E99">A21</font></i>: 
not <font color="Maroon" title="c15">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E22:97"><i><font color="Green" title="E96">A16</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E29:97"/>
<font color="Maroon" title="c15">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E13:97"><i><font color="Green" title="E89">A9</font></i></a>, <a class="txt" href="graphsp.html#E18:97"><i><font color="Green" title="E93">A13</font></i></a>, <a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></i>;<br/>
<a NAME="E30:97"/><b>then </b>
<font color="Maroon" title="c15">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c11">q</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E12:97"><i><font color="Green" title="E88">A8</font></i></a>, <a class="txt" href="graphsp.html#E15"><i><font color="Green" title="E15">Lm4</font></i></a></i>;<br/>
<a NAME="E31:97"/><b>then </b>
<font color="Maroon" title="c16">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c11">q</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E28:97"><i><font color="Green" title="E99">A21</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E32:97"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c16">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:97"><i><font color="Green" title="E81">A1</font></i></a>, <a class="txt" href="graphsp.html#E2:97"><i><font color="Green" title="E82">A2</font></i></a>, <a class="txt" href="graphsp.html#E7:97"><i><font color="Green" title="E84">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E33:97"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E13:97"><i><font color="Green" title="E89">A9</font></i></a>, <a class="txt" href="graphsp.html#E15:97"><i><font color="Green" title="E90">A10</font></i></a>, <a class="txt" href="graphsp.html#E18:97"><i><font color="Green" title="E93">A13</font></i></a>, <a class="txt" href="graphsp.html#E20:97"><i><font color="Green" title="E94">A14</font></i></a>, <a class="txt" href="graphsp.html#E27:97"><i><font color="Green" title="E98">A20</font></i></a>, <a class="ref" href="graphsp.html#T24" target="_self" title="GRAPHSP:th.24">Th24</a></i>;<br/>


</div>
