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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</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">f</font>, <font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c5">h</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">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="c8">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>set </b><font color="Maroon" title="c9">R</font> =  <a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font>;<br/>
<b>set </b><font color="Maroon" title="c10">M</font> =  <a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font>;<br/>
<b>set </b><font color="Maroon" title="c11">IN</font> =  <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>;<br/>
<b>set </b><font color="Maroon" title="c12">Ug</font> =  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>;<br/>
<b>assume </b><a NAME="E1:98"/><i><font color="Green" title="E82">A1</font></i>: 
( <font color="Maroon" title="c3">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="c7">W</font> &amp; <font color="Maroon" title="c8">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c3">f</font> &amp;  <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</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; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> )
 ;<br/>

<b>assume </b><a NAME="E2:98"/><i><font color="Green" title="E83">A2</font></i>: 
for <font color="Olive" title="b1">v3</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c6">G</font><br/> for <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="Olive" title="b1">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Olive" title="b1">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 holds <br/> ex <font color="Olive" title="b3">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b4">P</font> being <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> st <br/>( <font color="Olive" title="b3">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>,<font color="Olive" title="b2">j</font>,<font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b5">i</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="b5">i</font> &amp; <font color="Olive" title="b5">i</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="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> ) &amp; <font color="Olive" title="b4">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b4">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b4">P</font>,<font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b2">j</font></span>)</span> &amp; ( not <font color="Olive" title="b1">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> implies <font color="Olive" title="b4">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )
 ;<br/>

<b>assume </b><a NAME="E3:98"/><i><font color="Green" title="E84">A3</font></i>: 
( ( 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="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<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="b1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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; ( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) )
 ;<br/>

<b>set </b><font color="Maroon" title="c13">mi</font> = <span class="p1">(<span class="default"><span class="p2">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<b>set </b><font color="Maroon" title="c14">Ak</font> =  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>;<br/>
<a NAME="E4:98"/><i><font color="Green" title="E85">A4</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E5:98"/><i><font color="Green" title="E86">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="ref" href="graphsp.html#D20" target="_self" title="GRAPHSP:def.20">Def20</a></i>;<br/>
<a NAME="E6:98"/><i><font color="Green" title="E87">A6</font></i>: 
<span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graphsp.html#K2" title="GRAPHSP:func.2">:=</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="graphsp.html#D11" target="_self" title="GRAPHSP:def.11">Def11</a></i>;<br/>
<a NAME="E7:98"/><i><font color="Green" title="E88">A7</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="ref" href="graphsp.html#T34" target="_self" title="GRAPHSP:th.34">Th34</a></i>;<br/>
<a NAME="E8:98"/>
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="ref" href="graphsp.html#T23" target="_self" title="GRAPHSP:th.23">Th23</a></i>;<br/>
<a NAME="E9:98"/><b>then </b><i><font color="Green" title="E89">A8</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K18" title="GRAPHSP:func.18">Relax</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="graphsp.html#D15" target="_self" title="GRAPHSP:def.15">Def15</a></i>;<br/>
<a NAME="E10:98"/><i><font color="Green" title="E90">A9</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#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="ref" href="graphsp.html#D20" target="_self" title="GRAPHSP:def.20">Def20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">VG</font> = the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c6">G</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:98"/><i><font color="Green" title="E91">A10</font></i>: 
<font color="Maroon" title="c7">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>
 
<b>by </b><i><a class="ref" href="graph_5.html#D13" title="GRAPH_5:def.13">GRAPH_5:def 13</a></i>;<br/>
<a NAME="E13:98"/><b>then </b><i><font color="Green" title="E92">A11</font></i>: 
<font color="Maroon" title="c7">W</font> <a href="graph_5.html#R6" title="GRAPH_5:pred.6">is_weight_of</a> <font color="Maroon" title="c6">G</font>
 
<b>by </b><i><a class="ref" href="graph_5.html#T50" title="GRAPH_5:th.50">GRAPH_5:50</a></i>;<br/>
<a NAME="E14:98"/><i><font color="Green" title="E93">A12</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font>
 
;<br/>
<a NAME="E15:98"/><i><font color="Green" title="E94">A13</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="ref" href="graphsp.html#T42" target="_self" title="GRAPHSP:th.42">Th42</a></i>;<br/>
<a NAME="E16:98"/><i><font color="Green" title="E95">A14</font></i>: 
(  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> &amp;  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</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="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="ref" href="graphsp.html#T30" target="_self" title="GRAPHSP:th.30">Th30</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">Uh</font> =  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>;<br/>
<a NAME="E17:98"/><i><font color="Green" title="E96">A15</font></i>: 
(  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; not  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="ref" href="graphsp.html#T40" target="_self" title="GRAPHSP:th.40">Th40</a></i>;<br/>
<a NAME="E18:98"/><b>then </b><i><font color="Green" title="E97">A16</font></i>: 
 <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E19:98"/><i><font color="Green" title="E98">A17</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="graphsp.html#E52"><i><font color="Green" title="E46">Lm7</font></i></a></i>;<br/>
<a NAME="E20:98"/><i><font color="Green" title="E99">A18</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">h</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="ref" href="graphsp.html#T42" target="_self" title="GRAPHSP:th.42">Th42</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">vk</font> =  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c6">G</font> <b>by </b><i><a class="txt" href="graphsp.html#E10:98"><i><font color="Green" title="E90">A9</font></i></a>, <a class="txt" href="graphsp.html#E16:98"><i><font color="Green" title="E95">A14</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">pk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c19">PK</font> being  <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><b> such that </b><br/><a NAME="E23:98"/><i><font color="Green" title="E100">A19</font></i>: 
( <font color="Maroon" title="c18">pk</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>, <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">i</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">i</font> &amp; <font color="Olive" title="b1">i</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="c18">pk</font> holds <br/><font color="Maroon" title="c18">pk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> ) &amp; <font color="Maroon" title="c19">PK</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c18">pk</font> &amp; <font color="Maroon" title="c19">PK</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c17">vk</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c19">PK</font>,<font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> &amp; ( not <font color="Maroon" title="c17">vk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c19">PK</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E2:98"><i><font color="Green" title="E83">A2</font></i></a>, <a class="txt" href="graphsp.html#E16:98"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="graphsp.html#E17:98"><i><font color="Green" title="E96">A15</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">kk</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> <b> such that </b><br/><a NAME="E25:98"/><i><font color="Green" title="E101">A20</font></i>: 
( <font color="Maroon" title="c20">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c20">kk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">i</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="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c4">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">kk</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">i</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">i</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="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">kk</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">i</font></span>)</span> holds <br/><font color="Maroon" title="c20">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> ) )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="ref" href="graphsp.html#D10" target="_self" title="GRAPHSP:def.10">Def10</a></i>;<br/>
<b>set </b><font color="Maroon" title="c21">nAk</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>;<br/>
<a NAME="E26:98"/><i><font color="Green" title="E102">A21</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="graphsp.html#E16:98"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="graphsp.html#E82"><i><font color="Green" title="E71">Lm11</font></i></a></i>;<br/>
<a NAME="E27:98"/><b>then </b><i><font color="Green" title="E103">A22</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span> <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="c4">g</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E5:98"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="graphsp.html#E15:98"><i><font color="Green" title="E94">A13</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E28:98"/><i><font color="Green" title="E104">A23</font></i>: 
<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="graphsp.html#R2" title="GRAPHSP:pred.2">equal_at</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="ref" href="graphsp.html#T47" target="_self" title="GRAPHSP:th.47">Th47</a></i>;<br/>
<a NAME="E29:98"/>
<font color="Maroon" title="c19">PK</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c17">vk</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E23:98"><i><font color="Green" title="E100">A19</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="E30:98"/><b>then </b><i><font color="Green" title="E105">A24</font></i>: 
<font color="Maroon" title="c19">PK</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c17">vk</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/>
<a NAME="E31:98"/><b>then </b>
<font color="Maroon" title="c19">PK</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> 
 
<b>by </b><i><a class="ref" href="graph_5.html#D3" title="GRAPH_5:def.3">GRAPH_5:def 3</a></i>;<br/>
<a NAME="E32:98"/><b>then </b><i><font color="Green" title="E106">A25</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">PK</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></i>;<br/>
<a NAME="E33:98"/><i><font color="Green" title="E107">A26</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E4:98"><i><font color="Green" title="E85">A4</font></i></a>, <a class="txt" href="graphsp.html#E5:98"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="graphsp.html#E15:98"><i><font color="Green" title="E94">A13</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E108">A27</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E7:98"><i><font color="Green" title="E88">A7</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="graphsp.html#E6:98"><i><font color="Green" title="E87">A6</font></i></a>, <a class="txt" href="graphsp.html#E16:98"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="graphsp.html#E19:98"><i><font color="Green" title="E98">A17</font></i></a>, <a class="txt" href="graphsp.html#E33:98"><i><font color="Green" title="E107">A26</font></i></a>, <a class="ref" href="graphsp.html#T18" target="_self" title="GRAPHSP:th.18">Th18</a></i>
;<br/>
<i><font color="Green" title="E109">A28</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E7:98"><i><font color="Green" title="E88">A7</font></i></a>, <a class="txt" href="graphsp.html#E27:98"><i><font color="Green" title="E103">A22</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c19">PK</font>,<font color="Maroon" title="c7">W</font>
<b>by </b><i><a class="txt" href="graphsp.html#E6:98"><i><font color="Green" title="E87">A6</font></i></a>, <a class="txt" href="graphsp.html#E23:98"><i><font color="Green" title="E100">A19</font></i></a>, <a class="txt" href="graphsp.html#E26:98"><i><font color="Green" title="E102">A21</font></i></a>, <a class="txt" href="graphsp.html#E27:98"><i><font color="Green" title="E103">A22</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self" title="GRAPHSP:th.19">Th19</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c22">nk</font> = <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>;<br/>
<a NAME="E36:98"/><i><font color="Green" title="E110">A29</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span> &amp; <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="graphsp.html#E16:98"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="graphsp.html#E83"><i><font color="Green" title="E72">Lm12</font></i></a></i>;<br/>
<a NAME="E37:98"/>
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E16:98"><i><font color="Green" title="E95">A14</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E38:98"/><b>then </b><i><font color="Green" title="E111">A30</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E39:98"/><i><font color="Green" title="E112">A31</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span> <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="c4">g</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E5:98"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="graphsp.html#E15:98"><i><font color="Green" title="E94">A13</font></i></a>, <a class="txt" href="graphsp.html#E36:98"><i><font color="Green" title="E110">A29</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E40:98"/><b>then </b><i><font color="Green" title="E113">A32</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E36:98"><i><font color="Green" title="E110">A29</font></i></a>, <a class="txt" href="graphsp.html#E38:98"><i><font color="Green" title="E111">A30</font></i></a>, <a class="ref" href="graphsp.html#T32" target="_self" title="GRAPHSP:th.32">Th32</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_3"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c23">Wke</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span>;<br/><b>assume </b><a NAME="E1:98_3"/>
<span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c1">n</font>, <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 ;<br/><a NAME="E2:98_3"/><b>then </b><i><font color="Green" title="E114">A33</font></i>: 
( ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></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 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="graphsp.html#K17" title="GRAPHSP:func.17">newpathcost</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="graphsp.html#D13" target="_self" title="GRAPHSP:def.13">Def13</a></i>;<br/><a NAME="E3:98_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#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E4:98_3"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E34:98"><i><font color="Green" title="E108">A27</font></i></a>, <a class="txt" href="graphsp.html#E40:98"><i><font color="Green" title="E113">A32</font></i></a>, <a class="txt" href="graphsp.html#E2:98_3"><i><font color="Green" title="E114">A33</font></i></a>, <a class="ref" href="graphsp.html#T30" target="_self" title="GRAPHSP:th.30">Th30</a></i>;<br/></div><b>end;</b></div>
<a NAME="E42:98"/><b>then </b>
not <span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/>
<a NAME="E43:98"/><b>then </b><i><font color="Green" title="E114">A34</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E7:98"><i><font color="Green" title="E88">A7</font></i></a>, <a class="txt" href="graphsp.html#E9:98"><i><font color="Green" title="E89">A8</font></i></a>, <a class="txt" href="graphsp.html#E36:98"><i><font color="Green" title="E110">A29</font></i></a>, <a class="txt" href="graphsp.html#E38:98"><i><font color="Green" title="E111">A30</font></i></a>, <a class="txt" href="graphsp.html#E39:98"><i><font color="Green" title="E112">A31</font></i></a>, <a class="txt" href="graphsp.html#E40:98"><i><font color="Green" title="E113">A32</font></i></a>, <a class="ref" href="graphsp.html#D14" target="_self" title="GRAPHSP:def.14">Def14</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c23">v3</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>let </b><font color="Maroon" title="c24">j</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>assume </b><a NAME="E1:98_4"/><i><font color="Green" title="E115">A35</font></i>: 
( <font color="Maroon" title="c23">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Maroon" title="c23">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j</font> &amp; <font color="Maroon" title="c5">h</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 ;<br/><b>set </b><font color="Maroon" title="c25">nj</font> = <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font>;<br/><a NAME="E2:98_4"/>
<font color="Maroon" title="c24">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">VG</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98_4"><i><font color="Green" title="E115">A35</font></i></a></i>;<br/><a NAME="E3:98_4"/><b>then </b><i><font color="Green" title="E116">A36</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">j</font> &amp; <font color="Maroon" title="c24">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E10:98"><i><font color="Green" title="E90">A9</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:98_4"/><b>then </b><i><font color="Green" title="E117">A37</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> &amp; <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graphsp.html#E83"><i><font color="Green" title="E72">Lm12</font></i></a></i>;<br/><a NAME="E5:98_4"/><b>then </b><i><font color="Green" title="E118">A38</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</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="c4">g</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:98"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="graphsp.html#E15:98"><i><font color="Green" title="E94">A13</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:98_4"/><i><font color="Green" title="E119">A39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j</font>
 <b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E7:98_4"/>
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:98_4"><i><font color="Green" title="E116">A36</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E8:98_4"/><b>then </b><i><font color="Green" title="E120">A40</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>set </b><font color="Maroon" title="c26">m2</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font>;<br/><a NAME="E9:98_4"/><i><font color="Green" title="E121">A41</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E14:98"><i><font color="Green" title="E93">A12</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4"><i><font color="Green" title="E116">A36</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E10:98_4"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:98_4"><i><font color="Green" title="E116">A36</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E11:98_4"/><b>then </b><i><font color="Green" title="E122">A42</font></i>: 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E12:98_4"/><i><font color="Green" title="E123">A43</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j</font>
 <b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E13:98_4"/><i><font color="Green" title="E124">A44</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graphsp.html#E3:98_4"><i><font color="Green" title="E116">A36</font></i></a>, <a class="txt" href="graphsp.html#E82"><i><font color="Green" title="E71">Lm11</font></i></a></i>;<br/><a NAME="E14:98_4"/><b>then </b><i><font color="Green" title="E125">A45</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</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="c4">g</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:98"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="graphsp.html#E15:98"><i><font color="Green" title="E94">A13</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:98_4"/><i><font color="Green" title="E126">A46</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:98"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="graphsp.html#E7:98"><i><font color="Green" title="E88">A7</font></i></a>, <a class="txt" href="graphsp.html#E15:98"><i><font color="Green" title="E94">A13</font></i></a>, <a class="txt" href="graphsp.html#E13:98_4"><i><font color="Green" title="E124">A44</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:98_4"/><i><font color="Green" title="E127">A47</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E6:98"><i><font color="Green" title="E87">A6</font></i></a>, <a class="txt" href="graphsp.html#E16:98"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4"><i><font color="Green" title="E117">A37</font></i></a>, <a class="txt" href="graphsp.html#E5:98_4"><i><font color="Green" title="E118">A38</font></i></a>, <a class="txt" href="graphsp.html#E8:98_4"><i><font color="Green" title="E120">A40</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self" title="GRAPHSP:th.19">Th19</a></i>;<br/><a NAME="E17:98_4"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E51"><i><font color="Green" title="E45">Lm6</font></i></a></i>;<br/><a NAME="E18:98_4"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E11:98_4"><i><font color="Green" title="E122">A42</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E19:98_4"/><b>then </b><i><font color="Green" title="E128">A48</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E6:98"><i><font color="Green" title="E87">A6</font></i></a>, <a class="txt" href="graphsp.html#E16:98"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="graphsp.html#E13:98_4"><i><font color="Green" title="E124">A44</font></i></a>, <a class="txt" href="graphsp.html#E14:98_4"><i><font color="Green" title="E125">A45</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self" title="GRAPHSP:th.19">Th19</a></i>;<br/><a NAME="E20:98_4"/><i><font color="Green" title="E129">A49</font></i>: 
<font color="Maroon" title="c24">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E19:98"><i><font color="Green" title="E98">A17</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4"><i><font color="Green" title="E116">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E21:98_4"/><b>then </b><i><font color="Green" title="E130">A50</font></i>: 
<font color="Maroon" title="c24">j</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="c4">g</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:98"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="graphsp.html#E15:98"><i><font color="Green" title="E94">A13</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4"><i><font color="Green" title="E116">A36</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E22:98_4"/><i><font color="Green" title="E131">A51</font></i>: 
<font color="Maroon" title="c24">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">h</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:98"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="graphsp.html#E20:98"><i><font color="Green" title="E99">A18</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4"><i><font color="Green" title="E116">A36</font></i></a>, <a class="txt" href="graphsp.html#E20:98_4"><i><font color="Green" title="E129">A49</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>set </b><font color="Maroon" title="c27">Akj</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font>;<br/><a NAME="E23:98_4"/><i><font color="Green" title="E132">A52</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> &amp;  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graphsp.html#E16:98"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4"><i><font color="Green" title="E116">A36</font></i></a>, <a class="txt" href="graphsp.html#E84"><i><font color="Green" title="E73">Lm13</font></i></a></i>;<br/><a NAME="E24:98_4"/><b>then </b><i><font color="Green" title="E133">A53</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</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="c4">g</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:98"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="graphsp.html#E15:98"><i><font color="Green" title="E94">A13</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E25:98_4"/><i><font color="Green" title="E134">A54</font></i>: 
( <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="graphsp.html#E16:98"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4"><i><font color="Green" title="E116">A36</font></i></a>, <a class="txt" href="graphsp.html#E85"><i><font color="Green" title="E74">Lm14</font></i></a></i>;<br/><i><font color="Green" title="E135">A55</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c24">j</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p5">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E7:98"><i><font color="Green" title="E88">A7</font></i></a>, <a class="txt" href="graphsp.html#E34:98"><i><font color="Green" title="E108">A27</font></i></a>, <a class="txt" href="graphsp.html#E24:98_4"><i><font color="Green" title="E133">A53</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p5">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E6:98"><i><font color="Green" title="E87">A6</font></i></a>, <a class="txt" href="graphsp.html#E23:98_4"><i><font color="Green" title="E132">A52</font></i></a>, <a class="txt" href="graphsp.html#E24:98_4"><i><font color="Green" title="E133">A53</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self" title="GRAPHSP:th.19">Th19</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p5">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E15:98"><i><font color="Green" title="E94">A13</font></i></a>, <a class="txt" href="graphsp.html#E28:98"><i><font color="Green" title="E104">A23</font></i></a>, <a class="txt" href="graphsp.html#E24:98_4"><i><font color="Green" title="E133">A53</font></i></a>, <a class="txt" href="graphsp.html#E25:98_4"><i><font color="Green" title="E134">A54</font></i></a>, <a class="ref" href="graphsp.html#D16" target="_self" title="GRAPHSP:def.16">Def16</a></i>
<br/>.= 
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> <font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c23">v3</font>,<font color="Maroon" title="c7">W</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4"><i><font color="Green" title="E115">A35</font></i></a>, <a class="ref" href="graphsp.html#D20" target="_self" title="GRAPHSP:def.20">Def20</a></i>
;<br/><a NAME="E27:98_4"/><i><font color="Green" title="E136">A56</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E7:98"><i><font color="Green" title="E88">A7</font></i></a>, <a class="txt" href="graphsp.html#E14:98_4"><i><font color="Green" title="E125">A45</font></i></a>, <a class="txt" href="graphsp.html#E19:98_4"><i><font color="Green" title="E128">A48</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_4_2"/>
(  not <span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font> or <span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1"><b>suppose </b></a><a NAME="E1:98_4_2_1"/><i><font color="Green" title="E137">A57</font></i>: 
not <span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><a NAME="E2:98_4_2_1"/><b>then </b><i><font color="Green" title="E138">A58</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E7:98"><i><font color="Green" title="E88">A7</font></i></a>, <a class="txt" href="graphsp.html#E9:98"><i><font color="Green" title="E89">A8</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4"><i><font color="Green" title="E117">A37</font></i></a>, <a class="txt" href="graphsp.html#E5:98_4"><i><font color="Green" title="E118">A38</font></i></a>, <a class="txt" href="graphsp.html#E8:98_4"><i><font color="Green" title="E120">A40</font></i></a>, <a class="txt" href="graphsp.html#E16:98_4"><i><font color="Green" title="E127">A47</font></i></a>, <a class="ref" href="graphsp.html#D14" target="_self" title="GRAPHSP:def.14">Def14</a></i>;<br/><b>then consider </b><font color="Maroon" title="c28">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c29">P</font> being  <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><b> such that </b><br/><a NAME="E4:98_4_2_1"/><i><font color="Green" title="E139">A59</font></i>: 
( <font color="Maroon" title="c28">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c24">j</font>,<font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">i</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">i</font> &amp; <font color="Olive" title="b1">i</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="c28">p</font> holds <br/><font color="Maroon" title="c28">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> ) &amp; <font color="Maroon" title="c29">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c28">p</font> &amp; <font color="Maroon" title="c29">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c23">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c29">P</font>,<font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> &amp; ( not <font color="Maroon" title="c23">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c29">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )
 <b>by </b><i><a class="txt" href="graphsp.html#E2:98"><i><font color="Green" title="E83">A2</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4"><i><font color="Green" title="E115">A35</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c30">p</font> = <font color="Maroon" title="c28">p</font>;<br/><b>take </b><font color="Maroon" title="c31">P</font> = <font color="Maroon" title="c29">P</font>;<br/><b>thus </b><a NAME="E5:98_4_2_1"/>
<font color="Maroon" title="c30">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c24">j</font>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E5:98"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="graphsp.html#E2:98_4_2_1"><i><font color="Green" title="E138">A58</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4_2_1"><i><font color="Green" title="E139">A59</font></i></a>, <a class="txt" href="graphsp.html#E90"><i><font color="Green" title="E78">Lm17</font></i></a></i>;<br/><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c32">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>assume </b><a NAME="E1:98_4_2_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">i</font> &amp; <font color="Maroon" title="c32">i</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="c30">p</font> )
 ;<br/><a NAME="E2:98_4_2_1_1"/><b>then </b>
<font color="Maroon" title="c30">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E4:98_4_2_1"><i><font color="Green" title="E139">A59</font></i></a></i>;<br/><b>hence </b><a NAME="E3:98_4_2_1_1"/>
<font color="Maroon" title="c30">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E18:98"><i><font color="Green" title="E97">A16</font></i></a></i>;<br/></div>
<b>end;</b></div><b>thus </b><a NAME="E7:98_4_2_1"/>
<font color="Maroon" title="c31">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c30">p</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E4:98_4_2_1"><i><font color="Green" title="E139">A59</font></i></a></i>;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_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="E1:98_4_2_1_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c24">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c24">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:98_4_2_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c24">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ;<br/><div class="add"><a NAME="E2:98_4_2_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="graphsp.html#K18" title="GRAPHSP:func.18">Relax</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c24">j</font> <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#E7:98"><i><font color="Green" title="E88">A7</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4"><i><font color="Green" title="E116">A36</font></i></a>, <a class="txt" href="graphsp.html#E21:98_4"><i><font color="Green" title="E130">A50</font></i></a>, <a class="ref" href="graphsp.html#D14" target="_self" title="GRAPHSP:def.14">Def14</a></i>;<br/><a NAME="E3:98_4_2_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c24">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <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>  : ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">h</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">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="c5">h</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graphsp.html#E9:98"><i><font color="Green" title="E89">A8</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4"><i><font color="Green" title="E116">A36</font></i></a>, <a class="txt" href="graphsp.html#E22:98_4"><i><font color="Green" title="E131">A51</font></i></a></i>;<br/><a NAME="E4:98_4_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E140">A60</font></i>: 
( <font color="Maroon" title="c24">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c24">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="graphsp.html#E17:98"><i><font color="Green" title="E96">A15</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c32">Q</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="c33">v4</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="E1:98_4_2_1_2_1_1_1"/><i><font color="Green" title="E141">A61</font></i>: 
( not <font color="Maroon" title="c33">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c32">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c33">v4</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font> )
 ;<br/><a NAME="E2:98_4_2_1_2_1_1_1"/><i><font color="Green" title="E142">A62</font></i>: 
<font color="Maroon" title="c33">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">VG</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c34">j4</font> = <font color="Maroon" title="c33">v4</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="E4:98_4_2_1_2_1_1_1"/><i><font color="Green" title="E143">A63</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c34">j4</font> &amp; <font color="Maroon" title="c34">j4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E10:98"><i><font color="Green" title="E90">A9</font></i></a>, <a class="txt" href="graphsp.html#E2:98_4_2_1_2_1_1_1"><i><font color="Green" title="E142">A62</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:98_4_2_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E144">A64</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c34">j4</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E3:98"><i><font color="Green" title="E84">A3</font></i></a>, <a class="txt" href="graphsp.html#E12:98"><i><font color="Green" title="E91">A10</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_1_2_1_1_1"><i><font color="Green" title="E141">A61</font></i></a>, <a class="txt" href="graphsp.html#E92"><i><font color="Green" title="E80">Lm19</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c35">q</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c36">R</font> being  <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><b> such that </b><br/><a NAME="E7:98_4_2_1_2_1_1_1"/><i><font color="Green" title="E145">A65</font></i>: 
( <font color="Maroon" title="c35">q</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c34">j4</font>,<font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">i</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">i</font> &amp; <font color="Olive" title="b1">i</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="c35">q</font> holds <br/><font color="Maroon" title="c35">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> ) &amp; <font color="Maroon" title="c36">R</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c35">q</font> &amp; <font color="Maroon" title="c36">R</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c33">v4</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c36">R</font>,<font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c34">j4</font></span>)</span> &amp; ( not <font color="Maroon" title="c33">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c36">R</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E2:98"><i><font color="Green" title="E83">A2</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_1_2_1_1_1"><i><font color="Green" title="E141">A61</font></i></a></i>;<br/><a NAME="E8:98_4_2_1_2_1_1_1"/><i><font color="Green" title="E146">A66</font></i>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c36">R</font>,<font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c32">Q</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98_4_2_1_2_1_1_1"><i><font color="Green" title="E141">A61</font></i></a>, <a class="txt" href="graphsp.html#E7:98_4_2_1_2_1_1_1"><i><font color="Green" title="E145">A65</font></i></a>, <a class="ref" href="graphsp.html#T9" target="_self" title="GRAPHSP:th.9">Th9</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_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="E1:98_4_2_1_2_1_1_1_1"/>
( <font color="Maroon" title="c24">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c24">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> )
 </span><b>by </b><i><a class="txt" href="graphsp.html#E4:98_4_2_1_2_1_1"><i><font color="Green" title="E140">A60</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:98_4_2_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c24">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c37">PP</font> being  <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><b> such that </b><br/><a NAME="E3:98_4_2_1_2_1_1_1_1_1"/><i><font color="Green" title="E147">A67</font></i>: 
( <font color="Maroon" title="c37">PP</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c23">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c37">PP</font>,<font color="Maroon" title="c7">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c36">R</font>,<font color="Maroon" title="c7">W</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98_4"><i><font color="Green" title="E115">A35</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_1_2_1_1_1"><i><font color="Green" title="E141">A61</font></i></a>, <a class="txt" href="graphsp.html#E7:98_4_2_1_2_1_1_1"><i><font color="Green" title="E145">A65</font></i></a>, <a class="ref" href="graph_5.html#D19" title="GRAPH_5:def.19">GRAPH_5:def 19</a></i>;<br/><b>thus </b><a NAME="E4:98_4_2_1_2_1_1_1_1_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c31">P</font>,<font color="Maroon" title="c7">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c32">Q</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E4:98_4_2_1"><i><font color="Green" title="E139">A59</font></i></a>, <a class="txt" href="graphsp.html#E8:98_4_2_1_2_1_1_1"><i><font color="Green" title="E146">A66</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4_2_1_2_1_1_1_1_1"><i><font color="Green" title="E147">A67</font></i></a>, <a class="ref" href="graphsp.html#T9" target="_self" title="GRAPHSP:th.9">Th9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:98_4_2_1_2_1_1_1_1_2"/><i><font color="Green" title="E147">A68</font></i>: 
<font color="Maroon" title="c24">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><a NAME="E2:98_4_2_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c34">j4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E19:98"><i><font color="Green" title="E98">A17</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4_2_1_2_1_1_1"><i><font color="Green" title="E143">A63</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:98_4_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E148">A69</font></i>: 
<font color="Maroon" title="c34">j4</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="c4">g</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:98"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="graphsp.html#E15:98"><i><font color="Green" title="E94">A13</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4_2_1_2_1_1_1"><i><font color="Green" title="E143">A63</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:98_4_2_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c34">j4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98_4_2_1_2_1_1_1"><i><font color="Green" title="E141">A61</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4_2_1_2_1_1_1"><i><font color="Green" title="E143">A63</font></i></a></i>;<br/><a NAME="E5:98_4_2_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c34">j4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <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>  : ( <font color="Olive" title="b1">i</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="c4">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">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="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graphsp.html#E4:98_4_2_1_2_1_1_1"><i><font color="Green" title="E143">A63</font></i></a>, <a class="txt" href="graphsp.html#E5:98_4_2_1_2_1_1_1"><i><font color="Green" title="E144">A64</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4_2_1_2_1_1_1_1_2"><i><font color="Green" title="E148">A69</font></i></a></i>;<br/><a NAME="E6:98_4_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E149">A70</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c34">j4</font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E25:98"><i><font color="Green" title="E101">A20</font></i></a></i>;<br/><a NAME="E7:98_4_2_1_2_1_1_1_1_2"/><i><font color="Green" title="E150">A71</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E27:98"><i><font color="Green" title="E103">A22</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E8:98_4_2_1_2_1_1_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c34">j4</font> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c34">j4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graphsp.html#E4:98_4_2_1_2_1_1_1"><i><font color="Green" title="E143">A63</font></i></a>, <a class="txt" href="graphsp.html#E82"><i><font color="Green" title="E71">Lm11</font></i></a></i>;<br/><a NAME="E9:98_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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c34">j4</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="c4">g</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:98"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="graphsp.html#E15:98"><i><font color="Green" title="E94">A13</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E10:98_4_2_1_2_1_1_1_1_2"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c31">P</font>,<font color="Maroon" title="c7">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c32">Q</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E4:98_4_2_1"><i><font color="Green" title="E139">A59</font></i></a>, <a class="txt" href="graphsp.html#E7:98_4_2_1_2_1_1_1"><i><font color="Green" title="E145">A65</font></i></a>, <a class="txt" href="graphsp.html#E8:98_4_2_1_2_1_1_1"><i><font color="Green" title="E146">A66</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_1_2_1_1_1_1_2"><i><font color="Green" title="E147">A68</font></i></a>, <a class="txt" href="graphsp.html#E6:98_4_2_1_2_1_1_1_1_2"><i><font color="Green" title="E149">A70</font></i></a>, <a class="txt" href="graphsp.html#E7:98_4_2_1_2_1_1_1_1_2"><i><font color="Green" title="E150">A71</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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="E6:98_4_2_1_2_1_1"/>
<font color="Maroon" title="c31">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c23">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E12:98"><i><font color="Green" title="E91">A10</font></i></a>, <a class="txt" href="graphsp.html#E18:98"><i><font color="Green" title="E97">A16</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4"><i><font color="Green" title="E115">A35</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4_2_1"><i><font color="Green" title="E139">A59</font></i></a>, <a class="ref" href="graph_5.html#T68" title="GRAPH_5:th.68">GRAPH_5:68</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:98_4_2_1_2_1_2"/><i><font color="Green" title="E140">A72</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c24">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ;<br/><div class="add"><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_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="E1:98_4_2_1_2_1_2_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:98_4_2_1_2_1_2_1_1_1"/><i><font color="Green" title="E141">A73</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:98_4_2_1_2_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E142">A74</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graphsp.html#K17" title="GRAPHSP:func.17">newpathcost</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c24">j</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E6:98_4"><i><font color="Green" title="E119">A39</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_1"><i><font color="Green" title="E137">A57</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_1_2_1_2"><i><font color="Green" title="E140">A72</font></i></a>, <a class="ref" href="graphsp.html#D13" target="_self" title="GRAPHSP:def.13">Def13</a></i>;<br/><a NAME="E3:98_4_2_1_2_1_2_1_1_1"/><i><font color="Green" title="E143">A75</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c31">P</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E15:98_4"><i><font color="Green" title="E126">A46</font></i></a>, <a class="txt" href="graphsp.html#E19:98_4"><i><font color="Green" title="E128">A48</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4_2_1"><i><font color="Green" title="E139">A59</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c32">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:98_4_2_1_2_1_2_1_1_1"/><i><font color="Green" title="E144">A76</font></i>: 
( <font color="Maroon" title="c32">e</font> <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> &amp; <font color="Maroon" title="c32">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c23">v3</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E26:98_4"><i><font color="Green" title="E135">A55</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_1_2_1_2_1_1_1"><i><font color="Green" title="E141">A73</font></i></a>, <a class="ref" href="graphsp.html#T24" target="_self" title="GRAPHSP:th.24">Th24</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c33">pe</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c32">e</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as  <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> <b>by </b><i><a class="txt" href="graphsp.html#E5:98_4_2_1_2_1_2_1_1_1"><i><font color="Green" title="E144">A76</font></i></a>, <a class="ref" href="graphsp.html#T5" target="_self" title="GRAPHSP:th.5">Th5</a></i>;<br/><a NAME="E7:98_4_2_1_2_1_2_1_1_1"/><i><font color="Green" title="E145">A77</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">pe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c33">pe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">e</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><b>then consider </b><font color="Maroon" title="c34">Q</font> being  <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><b> such that </b><br/><a NAME="E9:98_4_2_1_2_1_2_1_1_1"/><i><font color="Green" title="E146">A78</font></i>: 
( <font color="Maroon" title="c34">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">PK</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">pe</font> &amp; <font color="Maroon" title="c34">Q</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c23">v3</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E30:98"><i><font color="Green" title="E105">A24</font></i></a>, <a class="txt" href="graphsp.html#E32:98"><i><font color="Green" title="E106">A25</font></i></a>, <a class="txt" href="graphsp.html#E5:98_4_2_1_2_1_2_1_1_1"><i><font color="Green" title="E144">A76</font></i></a>, <a class="ref" href="graph_5.html#T37" title="GRAPH_5:th.37">GRAPH_5:37</a></i>;<br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c33">pe</font>,<font color="Maroon" title="c7">W</font> = 
<font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">pe</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#E13:98"><i><font color="Green" title="E92">A11</font></i></a>, <a class="txt" href="graphsp.html#E7:98_4_2_1_2_1_2_1_1_1"><i><font color="Green" title="E145">A77</font></i></a>, <a class="ref" href="graphsp.html#T4" target="_self" title="GRAPHSP:th.4">Th4</a></i>
<br/>.= 
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> <font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c23">v3</font>,<font color="Maroon" title="c7">W</font>
<b>by </b><i><a class="txt" href="graphsp.html#E5:98_4_2_1_2_1_2_1_1_1"><i><font color="Green" title="E144">A76</font></i></a>, <a class="txt" href="graphsp.html#E7:98_4_2_1_2_1_2_1_1_1"><i><font color="Green" title="E145">A77</font></i></a>, <a class="ref" href="graphsp.html#T26" target="_self" title="GRAPHSP:th.26">Th26</a></i>
;<br/><a NAME="E11:98_4_2_1_2_1_2_1_1_1"/><b>then </b>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c34">Q</font>,<font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K17" title="GRAPHSP:func.17">newpathcost</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c24">j</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E13:98"><i><font color="Green" title="E92">A11</font></i></a>, <a class="txt" href="graphsp.html#E34:98"><i><font color="Green" title="E108">A27</font></i></a>, <a class="txt" href="graphsp.html#E35:98"><i><font color="Green" title="E109">A28</font></i></a>, <a class="txt" href="graphsp.html#E26:98_4"><i><font color="Green" title="E135">A55</font></i></a>, <a class="txt" href="graphsp.html#E9:98_4_2_1_2_1_2_1_1_1"><i><font color="Green" title="E146">A78</font></i></a>, <a class="ref" href="graph_5.html#T58" title="GRAPH_5:th.58">GRAPH_5:58</a></i>;<br/><b>hence </b><a NAME="E12:98_4_2_1_2_1_2_1_1_1"/>
<font color="Maroon" title="c31">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c23">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E12:98"><i><font color="Green" title="E91">A10</font></i></a>, <a class="txt" href="graphsp.html#E17:98"><i><font color="Green" title="E96">A15</font></i></a>, <a class="txt" href="graphsp.html#E23:98"><i><font color="Green" title="E100">A19</font></i></a>, <a class="txt" href="graphsp.html#E32:98"><i><font color="Green" title="E106">A25</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4"><i><font color="Green" title="E115">A35</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4_2_1"><i><font color="Green" title="E139">A59</font></i></a>, <a class="txt" href="graphsp.html#E2:98_4_2_1_2_1_2_1_1_1"><i><font color="Green" title="E142">A74</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4_2_1_2_1_2_1_1_1"><i><font color="Green" title="E143">A75</font></i></a>, <a class="txt" href="graphsp.html#E5:98_4_2_1_2_1_2_1_1_1"><i><font color="Green" title="E144">A76</font></i></a>, <a class="txt" href="graphsp.html#E9:98_4_2_1_2_1_2_1_1_1"><i><font color="Green" title="E146">A78</font></i></a>, <a class="ref" href="graph_5.html#T69" title="GRAPH_5:th.69">GRAPH_5:69</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:98_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#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:98_4_2_1_2_1_2_1_1_2"/><b>then </b>
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">e</font> <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> or  not <font color="Olive" title="b1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c23">v3</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E26:98_4"><i><font color="Green" title="E135">A55</font></i></a>, <a class="ref" href="graphsp.html#T24" target="_self" title="GRAPHSP:th.24">Th24</a></i>;<br/><b>hence </b><a NAME="E3:98_4_2_1_2_1_2_1_1_2"/>
<font color="Maroon" title="c31">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c23">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E12:98"><i><font color="Green" title="E91">A10</font></i></a>, <a class="txt" href="graphsp.html#E17:98"><i><font color="Green" title="E96">A15</font></i></a>, <a class="txt" href="graphsp.html#E23:98"><i><font color="Green" title="E100">A19</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4"><i><font color="Green" title="E115">A35</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4_2_1"><i><font color="Green" title="E139">A59</font></i></a>, <a class="ref" href="graphsp.html#T13" target="_self" title="GRAPHSP:th.13">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="E9:98_4_2_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c31">P</font>,<font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E9:98"><i><font color="Green" title="E89">A8</font></i></a>, <a class="txt" href="graphsp.html#E6:98_4"><i><font color="Green" title="E119">A39</font></i></a>, <a class="txt" href="graphsp.html#E9:98_4"><i><font color="Green" title="E121">A41</font></i></a>, <a class="txt" href="graphsp.html#E11:98_4"><i><font color="Green" title="E122">A42</font></i></a>, <a class="txt" href="graphsp.html#E12:98_4"><i><font color="Green" title="E123">A43</font></i></a>, <a class="txt" href="graphsp.html#E15:98_4"><i><font color="Green" title="E126">A46</font></i></a>, <a class="txt" href="graphsp.html#E19:98_4"><i><font color="Green" title="E128">A48</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_1"><i><font color="Green" title="E137">A57</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4_2_1"><i><font color="Green" title="E139">A59</font></i></a>, <a class="ref" href="graphsp.html#D14" target="_self" title="GRAPHSP:def.14">Def14</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:98_4_2_1_3"/><i><font color="Green" title="E140">A79</font></i>: 
not <font color="Maroon" title="c23">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>
 ;<br/><a NAME="E2:98_4_2_1_3"/><b>then </b><i><font color="Green" title="E141">A80</font></i>: 
not <font color="Maroon" title="c23">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E17:98"><i><font color="Green" title="E96">A15</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c32">v2</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="E1:98_4_2_1_3_1"/><i><font color="Green" title="E142">A81</font></i>: 
( <font color="Maroon" title="c32">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c32">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_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="E1:98_4_2_1_3_1_1"/>
( <font color="Maroon" title="c32">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c32">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> )
 </span><b>by </b><i><a class="txt" href="graphsp.html#E17:98"><i><font color="Green" title="E96">A15</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_1_3_1"><i><font color="Green" title="E142">A81</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:98_4_2_1_3_1_1_1"/>
<font color="Maroon" title="c32">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:98_4_2_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E143">A82</font></i>: 
<font color="Maroon" title="c32">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">vk</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>take </b><font color="Maroon" title="c33">PK</font> = <font color="Maroon" title="c19">PK</font>;<br/><b>thus </b><a NAME="E3:98_4_2_1_3_1_1_1"/>
<font color="Maroon" title="c33">PK</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c32">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E17:98"><i><font color="Green" title="E96">A15</font></i></a>, <a class="txt" href="graphsp.html#E23:98"><i><font color="Green" title="E100">A19</font></i></a>, <a class="txt" href="graphsp.html#E2:98_4_2_1_3_1_1_1"><i><font color="Green" title="E143">A82</font></i></a>, <a class="ref" href="graphsp.html#T8" target="_self" title="GRAPHSP:th.8">Th8</a></i>;<br/><a NAME="E4:98_4_2_1_3_1_1_1"/>
<font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c24">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E3:98_4"><i><font color="Green" title="E116">A36</font></i></a>, <a class="txt" href="graphsp.html#E21:98_4"><i><font color="Green" title="E130">A50</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4"><i><font color="Green" title="E115">A35</font></i></a>, <a class="txt" href="graphsp.html#E2:98_4_2_1_3"><i><font color="Green" title="E141">A80</font></i></a></i>;<br/><a NAME="E5:98_4_2_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c24">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <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>  : ( <font color="Olive" title="b1">i</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="c4">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">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="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98_4"><i><font color="Green" title="E115">A35</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4"><i><font color="Green" title="E116">A36</font></i></a>, <a class="txt" href="graphsp.html#E21:98_4"><i><font color="Green" title="E130">A50</font></i></a>, <a class="txt" href="graphsp.html#E2:98_4_2_1"><i><font color="Green" title="E138">A58</font></i></a></i>;<br/><a NAME="E6:98_4_2_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E144">A83</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E25:98"><i><font color="Green" title="E101">A20</font></i></a></i>;<br/><a NAME="E7:98_4_2_1_3_1_1_1"/>
<font color="Maroon" title="c4">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c33">PK</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E23:98"><i><font color="Green" title="E100">A19</font></i></a>, <a class="txt" href="graphsp.html#E27:98"><i><font color="Green" title="E103">A22</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E8:98_4_2_1_3_1_1_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c33">PK</font>,<font color="Maroon" title="c7">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c31">P</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E14:98_4"><i><font color="Green" title="E125">A45</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4_2_1"><i><font color="Green" title="E139">A59</font></i></a>, <a class="txt" href="graphsp.html#E6:98_4_2_1_3_1_1_1"><i><font color="Green" title="E144">A83</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:98_4_2_1_3_1_1_2"/><i><font color="Green" title="E143">A84</font></i>: 
<font color="Maroon" title="c32">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c33">Q</font> being  <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><b> such that </b><br/><a NAME="E3:98_4_2_1_3_1_1_2"/><i><font color="Green" title="E144">A85</font></i>: 
( <font color="Maroon" title="c33">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c32">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c33">Q</font>,<font color="Maroon" title="c7">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c31">P</font>,<font color="Maroon" title="c7">W</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E17:98"><i><font color="Green" title="E96">A15</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4_2_1"><i><font color="Green" title="E139">A59</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_1_3"><i><font color="Green" title="E140">A79</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_1_3_1"><i><font color="Green" title="E142">A81</font></i></a>, <a class="ref" href="graph_5.html#D19" title="GRAPH_5:def.19">GRAPH_5:def 19</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div><i><font color="Green" title="E145">A86</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_1_3_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c34">R</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="c35">v4</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="E1:98_4_2_1_3_1_1_2_1"/><i><font color="Green" title="E146">A87</font></i>: 
( not <font color="Maroon" title="c35">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c34">R</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c35">v4</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font> )
 ;<br/><a NAME="E2:98_4_2_1_3_1_1_2_1"/><i><font color="Green" title="E147">A88</font></i>: 
<font color="Maroon" title="c35">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">VG</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c36">j4</font> = <font color="Maroon" title="c35">v4</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="E4:98_4_2_1_3_1_1_2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">j4</font> &amp; <font color="Maroon" title="c36">j4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E10:98"><i><font color="Green" title="E90">A9</font></i></a>, <a class="txt" href="graphsp.html#E2:98_4_2_1_3_1_1_2_1"><i><font color="Green" title="E147">A88</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:98_4_2_1_3_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c36">j4</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E3:98"><i><font color="Green" title="E84">A3</font></i></a>, <a class="txt" href="graphsp.html#E12:98"><i><font color="Green" title="E91">A10</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_1_3_1_1_2_1"><i><font color="Green" title="E146">A87</font></i></a>, <a class="txt" href="graphsp.html#E92"><i><font color="Green" title="E80">Lm19</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c37">rn</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c38">RR</font> being  <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><b> such that </b><br/><a NAME="E7:98_4_2_1_3_1_1_2_1"/><i><font color="Green" title="E148">A89</font></i>: 
( <font color="Maroon" title="c37">rn</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c36">j4</font>,<font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">i</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">i</font> &amp; <font color="Olive" title="b1">i</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="c37">rn</font> holds <br/><font color="Maroon" title="c37">rn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> ) &amp; <font color="Maroon" title="c38">RR</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c37">rn</font> &amp; <font color="Maroon" title="c38">RR</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c35">v4</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c38">RR</font>,<font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c36">j4</font></span>)</span> &amp; ( not <font color="Maroon" title="c35">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c38">RR</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E2:98"><i><font color="Green" title="E83">A2</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_1_3_1_1_2_1"><i><font color="Green" title="E146">A87</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c39">QQ</font> being  <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><b> such that </b><br/><a NAME="E9:98_4_2_1_3_1_1_2_1"/><i><font color="Green" title="E149">A90</font></i>: 
( <font color="Maroon" title="c39">QQ</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c32">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c39">QQ</font>,<font color="Maroon" title="c7">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c38">RR</font>,<font color="Maroon" title="c7">W</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98_4_2_1_3_1"><i><font color="Green" title="E142">A81</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_1_3_1_1_2"><i><font color="Green" title="E143">A84</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_1_3_1_1_2_1"><i><font color="Green" title="E146">A87</font></i></a>, <a class="txt" href="graphsp.html#E7:98_4_2_1_3_1_1_2_1"><i><font color="Green" title="E148">A89</font></i></a>, <a class="ref" href="graph_5.html#D19" title="GRAPH_5:def.19">GRAPH_5:def 19</a></i>;<br/><a NAME="E10:98_4_2_1_3_1_1_2_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c39">QQ</font>,<font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c33">Q</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:98_4_2_1_3_1_1_2"><i><font color="Green" title="E144">A85</font></i></a>, <a class="txt" href="graphsp.html#E9:98_4_2_1_3_1_1_2_1"><i><font color="Green" title="E149">A90</font></i></a>, <a class="ref" href="graphsp.html#T9" target="_self" title="GRAPHSP:th.9">Th9</a></i>;<br/><b>hence </b><a NAME="E11:98_4_2_1_3_1_1_2_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c33">Q</font>,<font color="Maroon" title="c7">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c34">R</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98_4_2_1_3_1_1_2_1"><i><font color="Green" title="E146">A87</font></i></a>, <a class="txt" href="graphsp.html#E7:98_4_2_1_3_1_1_2_1"><i><font color="Green" title="E148">A89</font></i></a>, <a class="txt" href="graphsp.html#E9:98_4_2_1_3_1_1_2_1"><i><font color="Green" title="E149">A90</font></i></a>, <a class="ref" href="graphsp.html#T9" target="_self" title="GRAPHSP:th.9">Th9</a></i>;<br/></div><b>end;</b></div><b>take </b><font color="Maroon" title="c34">Q</font> = <font color="Maroon" title="c33">Q</font>;<br/><b>thus </b><a NAME="E5:98_4_2_1_3_1_1_2"/>
<font color="Maroon" title="c34">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c32">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E12:98"><i><font color="Green" title="E91">A10</font></i></a>, <a class="txt" href="graphsp.html#E18:98"><i><font color="Green" title="E97">A16</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_1_3_1"><i><font color="Green" title="E142">A81</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4_2_1_3_1_1_2"><i><font color="Green" title="E144">A85</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4_2_1_3_1_1_2"><i><font color="Green" title="E145">A86</font></i></a>, <a class="ref" href="graph_5.html#T68" title="GRAPH_5:th.68">GRAPH_5:68</a></i>;<br/><b>thus </b><a NAME="E6:98_4_2_1_3_1_1_2"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c34">Q</font>,<font color="Maroon" title="c7">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c31">P</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:98_4_2_1_3_1_1_2"><i><font color="Green" title="E144">A85</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="E4:98_4_2_1_3"/>
<font color="Maroon" title="c31">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="ref" href="graph_5.html#D19" title="GRAPH_5:def.19">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="98_4_2_2"><b>suppose </b></a><a NAME="E1:98_4_2_2"/><i><font color="Green" title="E137">A91</font></i>: 
<span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><a NAME="E2:98_4_2_2"/><b>then </b><i><font color="Green" title="E138">A92</font></i>: 
<span class="p1">(<span class="default"><a href="graphsp.html#K18" title="GRAPHSP:func.18">Relax</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E7:98"><i><font color="Green" title="E88">A7</font></i></a>, <a class="txt" href="graphsp.html#E34:98"><i><font color="Green" title="E108">A27</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4"><i><font color="Green" title="E117">A37</font></i></a>, <a class="txt" href="graphsp.html#E5:98_4"><i><font color="Green" title="E118">A38</font></i></a>, <a class="txt" href="graphsp.html#E8:98_4"><i><font color="Green" title="E120">A40</font></i></a>, <a class="ref" href="graphsp.html#D14" target="_self" title="GRAPHSP:def.14">Def14</a></i>;<br/><a NAME="E3:98_4_2_2"/><i><font color="Green" title="E139">A93</font></i>: 
( ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">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 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="graphsp.html#K17" title="GRAPHSP:func.17">newpathcost</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c24">j</font> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c24">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 <b>by </b><i><a class="txt" href="graphsp.html#E6:98_4"><i><font color="Green" title="E119">A39</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2"><i><font color="Green" title="E137">A91</font></i></a>, <a class="ref" href="graphsp.html#D13" target="_self" title="GRAPHSP:def.13">Def13</a></i>;<br/><a NAME="E4:98_4_2_2"/><i><font color="Green" title="E140">A94</font></i>: 
 <a href="graphsp.html#K17" title="GRAPHSP:func.17">newpathcost</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c24">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> <font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c23">v3</font>,<font color="Maroon" title="c7">W</font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E34:98"><i><font color="Green" title="E108">A27</font></i></a>, <a class="txt" href="graphsp.html#E26:98_4"><i><font color="Green" title="E135">A55</font></i></a></i>;<br/><div><i><font color="Green" title="E141">A95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:98_4_2_2_1"/><i><font color="Green" title="E142">A96</font></i>: 
 <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j</font>
 ;<br/><a NAME="E2:98_4_2_2_1"/><b>then </b><i><font color="Green" title="E143">A97</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E6:98"><i><font color="Green" title="E87">A6</font></i></a>, <a class="txt" href="graphsp.html#E16:98"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4"><i><font color="Green" title="E117">A37</font></i></a>, <a class="txt" href="graphsp.html#E5:98_4"><i><font color="Green" title="E118">A38</font></i></a>, <a class="txt" href="graphsp.html#E8:98_4"><i><font color="Green" title="E120">A40</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self" title="GRAPHSP:th.19">Th19</a></i>;<br/><a NAME="E3:98_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#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> <font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c23">v3</font>,<font color="Maroon" title="c7">W</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0
 <b>by </b><i><a class="txt" href="graphsp.html#E26:98_4"><i><font color="Green" title="E135">A55</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4_2_2"><i><font color="Green" title="E139">A93</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E4:98_4_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E6:98_4"><i><font color="Green" title="E119">A39</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2"><i><font color="Green" title="E137">A91</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4_2_2"><i><font color="Green" title="E140">A94</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2_1"><i><font color="Green" title="E142">A96</font></i></a>, <a class="txt" href="graphsp.html#E2:98_4_2_2_1"><i><font color="Green" title="E143">A97</font></i></a>, <a class="ref" href="graphsp.html#D13" target="_self" title="GRAPHSP:def.13">Def13</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E142">A98</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:98_4_2_2_2"/>
<font color="Maroon" title="c24">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c28">i</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> <b> such that </b><br/><a NAME="E3:98_4_2_2_2"/><i><font color="Green" title="E143">A99</font></i>: 
( <font color="Maroon" title="c24">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">i</font> &amp; <font color="Maroon" title="c28">i</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="c4">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">i</font> &amp; <font color="Maroon" title="c28">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="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c28">i</font> <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>thus </b><a NAME="E4:98_4_2_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E19:98"><i><font color="Green" title="E98">A17</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4_2_2"><i><font color="Green" title="E139">A93</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4_2_2_2"><i><font color="Green" title="E143">A99</font></i></a>, <a class="ref" href="graphsp.html#T33" target="_self" title="GRAPHSP:th.33">Th33</a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c28">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:98_4_2_2"/><i><font color="Green" title="E143">A100</font></i>: 
( <font color="Maroon" title="c28">e</font> <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> &amp; <font color="Maroon" title="c28">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c23">v3</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E26:98_4"><i><font color="Green" title="E135">A55</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4_2_2"><i><font color="Green" title="E139">A93</font></i></a>, <a class="ref" href="graphsp.html#T24" target="_self" title="GRAPHSP:th.24">Th24</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">pe</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c28">e</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as  <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> <b>by </b><i><a class="txt" href="graphsp.html#E8:98_4_2_2"><i><font color="Green" title="E143">A100</font></i></a>, <a class="ref" href="graphsp.html#T5" target="_self" title="GRAPHSP:th.5">Th5</a></i>;<br/><a NAME="E10:98_4_2_2"/><i><font color="Green" title="E144">A101</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">pe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c29">pe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">e</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><b>then consider </b><font color="Maroon" title="c30">Q</font> being  <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><b> such that </b><br/><a NAME="E12:98_4_2_2"/><i><font color="Green" title="E145">A102</font></i>: 
( <font color="Maroon" title="c30">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">PK</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">pe</font> &amp; <font color="Maroon" title="c30">Q</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c23">v3</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E30:98"><i><font color="Green" title="E105">A24</font></i></a>, <a class="txt" href="graphsp.html#E32:98"><i><font color="Green" title="E106">A25</font></i></a>, <a class="txt" href="graphsp.html#E8:98_4_2_2"><i><font color="Green" title="E143">A100</font></i></a>, <a class="ref" href="graph_5.html#T37" title="GRAPH_5:th.37">GRAPH_5:37</a></i>;<br/><b>take </b><font color="Maroon" title="c31">q</font> = <font color="Maroon" title="c18">pk</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c24">j</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>;<br/><b>take </b><font color="Maroon" title="c32">Q</font> = <font color="Maroon" title="c30">Q</font>;<br/><b>thus </b><a NAME="E13:98_4_2_2"/>
<font color="Maroon" title="c31">q</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c24">j</font>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E5:98"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="graphsp.html#E9:98"><i><font color="Green" title="E89">A8</font></i></a>, <a class="txt" href="graphsp.html#E23:98"><i><font color="Green" title="E100">A19</font></i></a>, <a class="txt" href="graphsp.html#E43:98"><i><font color="Green" title="E114">A34</font></i></a>, <a class="txt" href="graphsp.html#E2:98_4_2_2"><i><font color="Green" title="E138">A92</font></i></a>, <a class="txt" href="graphsp.html#E5:98_4_2_2"><i><font color="Green" title="E141">A95</font></i></a>, <a class="txt" href="graphsp.html#E6:98_4_2_2"><i><font color="Green" title="E142">A98</font></i></a>, <a class="txt" href="graphsp.html#E91"><i><font color="Green" title="E79">Lm18</font></i></a></i>;<br/><a NAME="E14:98_4_2_2"/><i><font color="Green" title="E146">A103</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pk</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c18">pk</font> <a href="graphsp.html#R3" title="GRAPHSP:pred.3">is_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>, <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E23:98"><i><font color="Green" title="E100">A19</font></i></a>, <a class="ref" href="graphsp.html#D18" target="_self" title="GRAPHSP:def.18">Def18</a></i>;<br/><b>then </b><i><font color="Green" title="E147">A104</font></i>: <font color="Maroon" title="c31">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="c18">pk</font></span>)</span> = 
<font color="Maroon" title="c18">pk</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="c18">pk</font></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E6"><i><font color="Green" title="E6">Lm1</font></i></a></i>
<br/>.= 
 <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="graphsp.html#E14:98_4_2_2"><i><font color="Green" title="E146">A103</font></i></a>, <a class="ref" href="graphsp.html#D17" target="_self" title="GRAPHSP:def.17">Def17</a></i>
;<br/><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c33">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>assume </b><a NAME="E1:98_4_2_2_4"/><i><font color="Green" title="E148">A105</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c33">i</font> &amp; <font color="Maroon" title="c33">i</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="c31">q</font> )
 ;<br/><a NAME="E2:98_4_2_2_4"/><b>then </b>
<font color="Maroon" title="c33">i</font> <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="c18">pk</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/><a NAME="E3:98_4_2_2_4"/><b>then </b><i><font color="Green" title="E149">A106</font></i>: 
<font color="Maroon" title="c33">i</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="c18">pk</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_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="E1:98_4_2_2_4_1_1"/>
( <font color="Maroon" title="c33">i</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="c18">pk</font> or <font color="Maroon" title="c33">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pk</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:98_4_2_2_4_1_1_1"/>
<font color="Maroon" title="c33">i</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="c18">pk</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:98_4_2_2_4_1_1_1"/>
( <font color="Maroon" title="c31">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c31">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E15:98_4_2_2"><i><font color="Green" title="E147">A104</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:98_4_2_2_4_1_1_2"/>
<font color="Maroon" title="c33">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pk</font>
 ;<br/><div class="add"><a NAME="E2:98_4_2_2_4_1_1_2"/><b>then </b><i><font color="Green" title="E150">A107</font></i>: 
<font color="Maroon" title="c33">i</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="c18">pk</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:98_4_2_2_4"><i><font color="Green" title="E149">A106</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:98_4_2_2_4_1_1_2"/>
<font color="Maroon" title="c31">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">pk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98_4_2_2_4"><i><font color="Green" title="E148">A105</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4_2_2_4"><i><font color="Green" title="E149">A106</font></i></a>, <a class="txt" href="graphsp.html#E6"><i><font color="Green" title="E6">Lm1</font></i></a></i>;<br/><b>hence </b><a NAME="E4:98_4_2_2_4_1_1_2"/>
( <font color="Maroon" title="c31">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c31">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E23:98"><i><font color="Green" title="E100">A19</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2_4"><i><font color="Green" title="E148">A105</font></i></a>, <a class="txt" href="graphsp.html#E2:98_4_2_2_4_1_1_2"><i><font color="Green" title="E150">A107</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="E5:98_4_2_2_4"/>
<font color="Maroon" title="c31">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E17:98"><i><font color="Green" title="E96">A15</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div>
<b>end;</b></div><a NAME="E17:98_4_2_2"/><i><font color="Green" title="E148">A108</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">Q</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="c19">PK</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E10:98_4_2_2"><i><font color="Green" title="E144">A101</font></i></a>, <a class="txt" href="graphsp.html#E12:98_4_2_2"><i><font color="Green" title="E145">A102</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E18:98_4_2_2"/><i><font color="Green" title="E149">A109</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pk</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="c19">PK</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E23:98"><i><font color="Green" title="E100">A19</font></i></a>, <a class="ref" href="graphsp.html#D19" target="_self" title="GRAPHSP:def.19">Def19</a></i>;<br/><a NAME="E19:98_4_2_2"/><b>then </b><i><font color="Green" title="E150">A110</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c31">q</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="c32">Q</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E17:98_4_2_2"><i><font color="Green" title="E148">A108</font></i></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/><b>set </b><font color="Maroon" title="c33">FS</font> = the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c6">G</font>;<br/><b>set </b><font color="Maroon" title="c34">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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c35">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>assume </b><a NAME="E1:98_4_2_2_5"/><i><font color="Green" title="E151">A111</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">i</font> &amp; <font color="Maroon" title="c35">i</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="c32">Q</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_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="E1:98_4_2_2_5_1"/>
( <font color="Maroon" title="c35">i</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="c32">Q</font> or <font color="Maroon" title="c35">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">Q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_5_1_1"><b>suppose </b></a><a NAME="E1:98_4_2_2_5_1_1"/><i><font color="Green" title="E152">A112</font></i>: 
<font color="Maroon" title="c35">i</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="c32">Q</font>
 ;<br/><div class="add"><a NAME="E2:98_4_2_2_5_1_1"/><b>then </b><i><font color="Green" title="E153">A113</font></i>: 
<font color="Maroon" title="c32">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">e</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E10:98_4_2_2"><i><font color="Green" title="E144">A101</font></i></a>, <a class="txt" href="graphsp.html#E12:98_4_2_2"><i><font color="Green" title="E145">A102</font></i></a>, <a class="txt" href="graphsp.html#E17:98_4_2_2"><i><font color="Green" title="E148">A108</font></i></a>, <a class="txt" href="graphsp.html#E7"><i><font color="Green" title="E7">Lm2</font></i></a></i>;<br/><a NAME="E3:98_4_2_2_5_1_1"/><b>then </b><i><font color="Green" title="E154">A114</font></i>: 
( 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="c32">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">vk</font> &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="c32">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">v3</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E8:98_4_2_2"><i><font color="Green" title="E143">A100</font></i></a>, <a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></i>;<br/><b>thus </b><a NAME="E4:98_4_2_2_5_1_1"/>
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="c32">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E8:98_4_2_2"><i><font color="Green" title="E143">A100</font></i></a>, <a class="txt" href="graphsp.html#E15:98_4_2_2"><i><font color="Green" title="E147">A104</font></i></a>, <a class="txt" href="graphsp.html#E17:98_4_2_2"><i><font color="Green" title="E148">A108</font></i></a>, <a class="txt" href="graphsp.html#E18:98_4_2_2"><i><font color="Green" title="E149">A109</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2_5_1_1"><i><font color="Green" title="E152">A112</font></i></a>, <a class="txt" href="graphsp.html#E2:98_4_2_2_5_1_1"><i><font color="Green" title="E153">A113</font></i></a>, <a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></i>;<br/><b>thus </b><a NAME="E5:98_4_2_2_5_1_1"/>
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="c32">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98_4"><i><font color="Green" title="E115">A35</font></i></a>, <a class="txt" href="graphsp.html#E17:98_4_2_2"><i><font color="Green" title="E148">A108</font></i></a>, <a class="txt" href="graphsp.html#E18:98_4_2_2"><i><font color="Green" title="E149">A109</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2_5_1_1"><i><font color="Green" title="E152">A112</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4_2_2_5_1_1"><i><font color="Green" title="E154">A114</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_5_1_2"><b>suppose </b></a><a NAME="E1:98_4_2_2_5_1_2"/>
<font color="Maroon" title="c35">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">Q</font>
 ;<br/><div class="add"><a NAME="E2:98_4_2_2_5_1_2"/><b>then </b><i><font color="Green" title="E152">A115</font></i>: 
<font color="Maroon" title="c35">i</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="c32">Q</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98_4_2_2_5"><i><font color="Green" title="E151">A111</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:98_4_2_2_5_1_2"/><b>then </b><i><font color="Green" title="E153">A116</font></i>: 
<font color="Maroon" title="c35">i</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="c19">PK</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E17:98_4_2_2"><i><font color="Green" title="E148">A108</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:98_4_2_2_5_1_2"/><b>then </b><i><font color="Green" title="E154">A117</font></i>: 
( 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="c19">PK</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">pk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font> &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="c19">PK</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">pk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="graphsp.html#E23:98"><i><font color="Green" title="E100">A19</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2_5"><i><font color="Green" title="E151">A111</font></i></a>, <a class="ref" href="graphsp.html#D19" target="_self" title="GRAPHSP:def.19">Def19</a></i>;<br/><a NAME="E5:98_4_2_2_5_1_2"/><i><font color="Green" title="E155">A118</font></i>: 
<font color="Maroon" title="c32">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">PK</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E12:98_4_2_2"><i><font color="Green" title="E145">A102</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2_5"><i><font color="Green" title="E151">A111</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4_2_2_5_1_2"><i><font color="Green" title="E153">A116</font></i></a>, <a class="txt" href="graphsp.html#E6"><i><font color="Green" title="E6">Lm1</font></i></a></i>;<br/><a NAME="E6:98_4_2_2_5_1_2"/><i><font color="Green" title="E156">A119</font></i>: 
<font color="Maroon" title="c35">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pk</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E17:98_4_2_2"><i><font color="Green" title="E148">A108</font></i></a>, <a class="txt" href="graphsp.html#E18:98_4_2_2"><i><font color="Green" title="E149">A109</font></i></a>, <a class="txt" href="graphsp.html#E2:98_4_2_2_5_1_2"><i><font color="Green" title="E152">A115</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>thus </b><a NAME="E7:98_4_2_2_5_1_2"/>
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="c32">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E17:98_4_2_2"><i><font color="Green" title="E148">A108</font></i></a>, <a class="txt" href="graphsp.html#E18:98_4_2_2"><i><font color="Green" title="E149">A109</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2_5"><i><font color="Green" title="E151">A111</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4_2_2_5_1_2"><i><font color="Green" title="E154">A117</font></i></a>, <a class="txt" href="graphsp.html#E5:98_4_2_2_5_1_2"><i><font color="Green" title="E155">A118</font></i></a>, <a class="txt" href="graphsp.html#E6"><i><font color="Green" title="E6">Lm1</font></i></a></i>;<br/><a NAME="E8:98_4_2_2_5_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><b>hence </b><a NAME="E9:98_4_2_2_5_1_2"/>
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="c32">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E4:98_4_2_2_5_1_2"><i><font color="Green" title="E154">A117</font></i></a>, <a class="txt" href="graphsp.html#E5:98_4_2_2_5_1_2"><i><font color="Green" title="E155">A118</font></i></a>, <a class="txt" href="graphsp.html#E6:98_4_2_2_5_1_2"><i><font color="Green" title="E156">A119</font></i></a>, <a class="txt" href="graphsp.html#E6"><i><font color="Green" title="E6">Lm1</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="E21:98_4_2_2"/>
<font color="Maroon" title="c32">Q</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c31">q</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E19:98_4_2_2"><i><font color="Green" title="E150">A110</font></i></a>, <a class="ref" href="graphsp.html#D19" target="_self" title="GRAPHSP:def.19">Def19</a></i>;<br/><a NAME="E22:98_4_2_2"/><i><font color="Green" title="E151">A120</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c19">PK</font>,<font color="Maroon" title="c7">W</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c29">pe</font>,<font color="Maroon" title="c7">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c32">Q</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E13:98"><i><font color="Green" title="E92">A11</font></i></a>, <a class="txt" href="graphsp.html#E12:98_4_2_2"><i><font color="Green" title="E145">A102</font></i></a>, <a class="ref" href="graph_5.html#T58" title="GRAPH_5:th.58">GRAPH_5:58</a></i>;<br/><i><font color="Green" title="E152">A121</font></i>:  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c29">pe</font>,<font color="Maroon" title="c7">W</font> = 
<font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">pe</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#E13:98"><i><font color="Green" title="E92">A11</font></i></a>, <a class="txt" href="graphsp.html#E10:98_4_2_2"><i><font color="Green" title="E144">A101</font></i></a>, <a class="ref" href="graphsp.html#T4" target="_self" title="GRAPHSP:th.4">Th4</a></i>
<br/>.= 
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> <font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c23">v3</font>,<font color="Maroon" title="c7">W</font>
<b>by </b><i><a class="txt" href="graphsp.html#E8:98_4_2_2"><i><font color="Green" title="E143">A100</font></i></a>, <a class="txt" href="graphsp.html#E10:98_4_2_2"><i><font color="Green" title="E144">A101</font></i></a>, <a class="ref" href="graphsp.html#T26" target="_self" title="GRAPHSP:th.26">Th26</a></i>
;<br/><a NAME="E24:98_4_2_2"/><b>then </b><i><font color="Green" title="E153">A122</font></i>: 
 <a href="graphsp.html#K17" title="GRAPHSP:func.17">newpathcost</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c24">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c32">Q</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E13:98"><i><font color="Green" title="E92">A11</font></i></a>, <a class="txt" href="graphsp.html#E34:98"><i><font color="Green" title="E108">A27</font></i></a>, <a class="txt" href="graphsp.html#E35:98"><i><font color="Green" title="E109">A28</font></i></a>, <a class="txt" href="graphsp.html#E26:98_4"><i><font color="Green" title="E135">A55</font></i></a>, <a class="txt" href="graphsp.html#E12:98_4_2_2"><i><font color="Green" title="E145">A102</font></i></a>, <a class="ref" href="graph_5.html#T58" title="GRAPH_5:th.58">GRAPH_5:58</a></i>;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_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="E1:98_4_2_2_7_1"/>
( <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">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 or <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_7_1_1"><b>suppose </b></a><a NAME="E1:98_4_2_2_7_1_1"/><i><font color="Green" title="E154">A123</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">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/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_2_7_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c35">v2</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="E1:98_4_2_2_7_1_1_1"/><i><font color="Green" title="E155">A124</font></i>: 
<font color="Maroon" title="c35">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c36">m</font> = <font color="Maroon" title="c35">v2</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 href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 = 
<font color="Maroon" title="c3">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<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="c36">m</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E3:98"><i><font color="Green" title="E84">A3</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4"><i><font color="Green" title="E116">A36</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2_7_1_1"><i><font color="Green" title="E154">A123</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2_7_1_1_1"><i><font color="Green" title="E155">A124</font></i></a></i>
<br/>.= 
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> <font color="Maroon" title="c35">v2</font>,<font color="Maroon" title="c23">v3</font>,<font color="Maroon" title="c7">W</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4"><i><font color="Green" title="E115">A35</font></i></a>, <a class="ref" href="graphsp.html#D20" target="_self" title="GRAPHSP:def.20">Def20</a></i>
;<br/><b>hence </b><a NAME="E4:98_4_2_2_7_1_1_1"/>
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">e</font> <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> or  not <font color="Olive" title="b1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c35">v2</font>,<font color="Maroon" title="c23">v3</font> )
 <b>by </b><i><a class="ref" href="graphsp.html#T24" target="_self" title="GRAPHSP:th.24">Th24</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:98_4_2_2_7_1_1"/>
<font color="Maroon" title="c32">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c23">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E17:98"><i><font color="Green" title="E96">A15</font></i></a>, <a class="txt" href="graphsp.html#E23:98"><i><font color="Green" title="E100">A19</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4"><i><font color="Green" title="E115">A35</font></i></a>, <a class="txt" href="graphsp.html#E8:98_4_2_2"><i><font color="Green" title="E143">A100</font></i></a>, <a class="txt" href="graphsp.html#E12:98_4_2_2"><i><font color="Green" title="E145">A102</font></i></a>, <a class="ref" href="graphsp.html#T15" target="_self" title="GRAPHSP:th.15">Th15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_7_1_2"><b>suppose </b></a><a NAME="E1:98_4_2_2_7_1_2"/><i><font color="Green" title="E154">A125</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c35">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c36">P</font> being  <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><b> such that </b><br/><a NAME="E3:98_4_2_2_7_1_2"/><i><font color="Green" title="E155">A126</font></i>: 
( <font color="Maroon" title="c35">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c24">j</font>,<font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">i</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">i</font> &amp; <font color="Olive" title="b1">i</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="c35">p</font> holds <br/><font color="Maroon" title="c35">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> ) &amp; <font color="Maroon" title="c36">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c35">p</font> &amp; <font color="Maroon" title="c36">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c23">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c36">P</font>,<font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> &amp; ( not <font color="Maroon" title="c23">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c36">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )
 <b>by </b><i><a class="txt" href="graphsp.html#E2:98"><i><font color="Green" title="E83">A2</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4"><i><font color="Green" title="E115">A35</font></i></a></i>;<br/><b>thus </b><a NAME="E4:98_4_2_2_7_1_2"/>
<font color="Maroon" title="c32">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c23">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E6:98"><i><font color="Green" title="E87">A6</font></i></a>, <a class="txt" href="graphsp.html#E12:98"><i><font color="Green" title="E91">A10</font></i></a>, <a class="txt" href="graphsp.html#E16:98"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="graphsp.html#E17:98"><i><font color="Green" title="E96">A15</font></i></a>, <a class="txt" href="graphsp.html#E23:98"><i><font color="Green" title="E100">A19</font></i></a>, <a class="txt" href="graphsp.html#E32:98"><i><font color="Green" title="E106">A25</font></i></a>, <a class="txt" href="graphsp.html#E34:98"><i><font color="Green" title="E108">A27</font></i></a>, <a class="txt" href="graphsp.html#E35:98"><i><font color="Green" title="E109">A28</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4"><i><font color="Green" title="E115">A35</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4"><i><font color="Green" title="E117">A37</font></i></a>, <a class="txt" href="graphsp.html#E5:98_4"><i><font color="Green" title="E118">A38</font></i></a>, <a class="txt" href="graphsp.html#E8:98_4"><i><font color="Green" title="E120">A40</font></i></a>, <a class="txt" href="graphsp.html#E26:98_4"><i><font color="Green" title="E135">A55</font></i></a>, <a class="txt" href="graphsp.html#E27:98_4"><i><font color="Green" title="E136">A56</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4_2_2"><i><font color="Green" title="E139">A93</font></i></a>, <a class="txt" href="graphsp.html#E8:98_4_2_2"><i><font color="Green" title="E143">A100</font></i></a>, <a class="txt" href="graphsp.html#E12:98_4_2_2"><i><font color="Green" title="E145">A102</font></i></a>, <a class="txt" href="graphsp.html#E22:98_4_2_2"><i><font color="Green" title="E151">A120</font></i></a>, <a class="txt" href="graphsp.html#E23:98_4_2_2"><i><font color="Green" title="E152">A121</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2_7_1_2"><i><font color="Green" title="E154">A125</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4_2_2_7_1_2"><i><font color="Green" title="E155">A126</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self" title="GRAPHSP:th.19">Th19</a>, <a class="ref" href="graph_5.html#T69" title="GRAPH_5:th.69">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="E26:98_4_2_2"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c32">Q</font>,<font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E9:98"><i><font color="Green" title="E89">A8</font></i></a>, <a class="txt" href="graphsp.html#E6:98_4"><i><font color="Green" title="E119">A39</font></i></a>, <a class="txt" href="graphsp.html#E9:98_4"><i><font color="Green" title="E121">A41</font></i></a>, <a class="txt" href="graphsp.html#E11:98_4"><i><font color="Green" title="E122">A42</font></i></a>, <a class="txt" href="graphsp.html#E12:98_4"><i><font color="Green" title="E123">A43</font></i></a>, <a class="txt" href="graphsp.html#E15:98_4"><i><font color="Green" title="E126">A46</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2"><i><font color="Green" title="E137">A91</font></i></a>, <a class="txt" href="graphsp.html#E24:98_4_2_2"><i><font color="Green" title="E153">A122</font></i></a>, <a class="ref" href="graphsp.html#D14" target="_self" title="GRAPHSP:def.14">Def14</a></i>;<br/><a NAME="E27:98_4_2_2"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c29">pe</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E12:98"><i><font color="Green" title="E91">A10</font></i></a>, <a class="ref" href="graph_5.html#T54" title="GRAPH_5:th.54">GRAPH_5:54</a></i>;<br/><a NAME="E28:98_4_2_2"/><b>then </b><i><font color="Green" title="E154">A127</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c19">PK</font>,<font color="Maroon" title="c7">W</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c32">Q</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E22:98_4_2_2"><i><font color="Green" title="E151">A120</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:98_4_2_2_8"/>
not <font color="Maroon" title="c23">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_2_8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c35">v2</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="E1:98_4_2_2_8_1"/><i><font color="Green" title="E155">A128</font></i>: 
( <font color="Maroon" title="c35">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c35">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_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="E1:98_4_2_2_8_1_1"/>
( <font color="Maroon" title="c35">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c35">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> )
 </span><b>by </b><i><a class="txt" href="graphsp.html#E17:98"><i><font color="Green" title="E96">A15</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2_8_1"><i><font color="Green" title="E155">A128</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_8_1_1_1"><b>suppose </b></a><a NAME="E1:98_4_2_2_8_1_1_1"/>
<font color="Maroon" title="c35">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:98_4_2_2_8_1_1_1"/><b>then </b><i><font color="Green" title="E156">A129</font></i>: 
<font color="Maroon" title="c35">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>take </b><font color="Maroon" title="c36">PK</font> = <font color="Maroon" title="c19">PK</font>;<br/><b>thus </b><a NAME="E3:98_4_2_2_8_1_1_1"/>
<font color="Maroon" title="c36">PK</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c35">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E17:98"><i><font color="Green" title="E96">A15</font></i></a>, <a class="txt" href="graphsp.html#E23:98"><i><font color="Green" title="E100">A19</font></i></a>, <a class="txt" href="graphsp.html#E2:98_4_2_2_8_1_1_1"><i><font color="Green" title="E156">A129</font></i></a>, <a class="ref" href="graphsp.html#T8" target="_self" title="GRAPHSP:th.8">Th8</a></i>;<br/><b>thus </b><a NAME="E4:98_4_2_2_8_1_1_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c36">PK</font>,<font color="Maroon" title="c7">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c32">Q</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E28:98_4_2_2"><i><font color="Green" title="E154">A127</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_8_1_1_2"><b>suppose </b></a><a NAME="E1:98_4_2_2_8_1_1_2"/><i><font color="Green" title="E156">A130</font></i>: 
<font color="Maroon" title="c35">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c36">P</font> being  <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><b> such that </b><br/><a NAME="E3:98_4_2_2_8_1_1_2"/><i><font color="Green" title="E157">A131</font></i>: 
( <font color="Maroon" title="c36">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c35">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c36">P</font>,<font color="Maroon" title="c7">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c19">PK</font>,<font color="Maroon" title="c7">W</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E23:98"><i><font color="Green" title="E100">A19</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2_8_1"><i><font color="Green" title="E155">A128</font></i></a>, <a class="ref" href="graphsp.html#T40" target="_self" title="GRAPHSP:th.40">Th40</a>, <a class="ref" href="graph_5.html#D19" title="GRAPH_5:def.19">GRAPH_5:def 19</a></i>;<br/><div><i><font color="Green" title="E158">A132</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_2_8_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c37">R</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="c38">v4</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="E1:98_4_2_2_8_1_1_2_1"/><i><font color="Green" title="E159">A133</font></i>: 
( not <font color="Maroon" title="c38">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c37">R</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c38">v4</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font> )
 ;<br/><a NAME="E2:98_4_2_2_8_1_1_2_1"/><i><font color="Green" title="E160">A134</font></i>: 
<font color="Maroon" title="c38">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">VG</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c39">j4</font> = <font color="Maroon" title="c38">v4</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="E4:98_4_2_2_8_1_1_2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">j4</font> &amp; <font color="Maroon" title="c39">j4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E10:98"><i><font color="Green" title="E90">A9</font></i></a>, <a class="txt" href="graphsp.html#E2:98_4_2_2_8_1_1_2_1"><i><font color="Green" title="E160">A134</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:98_4_2_2_8_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c39">j4</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E3:98"><i><font color="Green" title="E84">A3</font></i></a>, <a class="txt" href="graphsp.html#E12:98"><i><font color="Green" title="E91">A10</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2_8_1_1_2_1"><i><font color="Green" title="E159">A133</font></i></a>, <a class="txt" href="graphsp.html#E92"><i><font color="Green" title="E80">Lm19</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c40">rn</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c41">RR</font> being  <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><b> such that </b><br/><a NAME="E7:98_4_2_2_8_1_1_2_1"/><i><font color="Green" title="E161">A135</font></i>: 
( <font color="Maroon" title="c40">rn</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c39">j4</font>,<font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">i</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">i</font> &amp; <font color="Olive" title="b1">i</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="c40">rn</font> holds <br/><font color="Maroon" title="c40">rn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> ) &amp; <font color="Maroon" title="c41">RR</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c40">rn</font> &amp; <font color="Maroon" title="c41">RR</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c38">v4</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c41">RR</font>,<font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c39">j4</font></span>)</span> &amp; ( not <font color="Maroon" title="c38">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c41">RR</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E2:98"><i><font color="Green" title="E83">A2</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2_8_1_1_2_1"><i><font color="Green" title="E159">A133</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c42">PP</font> being  <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><b> such that </b><br/><a NAME="E9:98_4_2_2_8_1_1_2_1"/><i><font color="Green" title="E162">A136</font></i>: 
( <font color="Maroon" title="c42">PP</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c35">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c42">PP</font>,<font color="Maroon" title="c7">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c41">RR</font>,<font color="Maroon" title="c7">W</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98_4_2_2_8_1"><i><font color="Green" title="E155">A128</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2_8_1_1_2"><i><font color="Green" title="E156">A130</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2_8_1_1_2_1"><i><font color="Green" title="E159">A133</font></i></a>, <a class="txt" href="graphsp.html#E7:98_4_2_2_8_1_1_2_1"><i><font color="Green" title="E161">A135</font></i></a>, <a class="ref" href="graph_5.html#D19" title="GRAPH_5:def.19">GRAPH_5:def 19</a></i>;<br/><a NAME="E10:98_4_2_2_8_1_1_2_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c42">PP</font>,<font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c36">P</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:98_4_2_2_8_1_1_2"><i><font color="Green" title="E157">A131</font></i></a>, <a class="txt" href="graphsp.html#E9:98_4_2_2_8_1_1_2_1"><i><font color="Green" title="E162">A136</font></i></a>, <a class="ref" href="graphsp.html#T9" target="_self" title="GRAPHSP:th.9">Th9</a></i>;<br/><b>hence </b><a NAME="E11:98_4_2_2_8_1_1_2_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c36">P</font>,<font color="Maroon" title="c7">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c37">R</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98_4_2_2_8_1_1_2_1"><i><font color="Green" title="E159">A133</font></i></a>, <a class="txt" href="graphsp.html#E7:98_4_2_2_8_1_1_2_1"><i><font color="Green" title="E161">A135</font></i></a>, <a class="txt" href="graphsp.html#E9:98_4_2_2_8_1_1_2_1"><i><font color="Green" title="E162">A136</font></i></a>, <a class="ref" href="graphsp.html#T9" target="_self" title="GRAPHSP:th.9">Th9</a></i>;<br/></div><b>end;</b></div><b>take </b><font color="Maroon" title="c37">P</font> = <font color="Maroon" title="c36">P</font>;<br/><b>thus </b><a NAME="E5:98_4_2_2_8_1_1_2"/>
<font color="Maroon" title="c37">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c35">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E12:98"><i><font color="Green" title="E91">A10</font></i></a>, <a class="txt" href="graphsp.html#E18:98"><i><font color="Green" title="E97">A16</font></i></a>, <a class="txt" href="graphsp.html#E1:98_4_2_2_8_1"><i><font color="Green" title="E155">A128</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4_2_2_8_1_1_2"><i><font color="Green" title="E157">A131</font></i></a>, <a class="txt" href="graphsp.html#E4:98_4_2_2_8_1_1_2"><i><font color="Green" title="E158">A132</font></i></a>, <a class="ref" href="graph_5.html#T68" title="GRAPH_5:th.68">GRAPH_5:68</a></i>;<br/><b>thus </b><a NAME="E6:98_4_2_2_8_1_1_2"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c37">P</font>,<font color="Maroon" title="c7">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c32">Q</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E28:98_4_2_2"><i><font color="Green" title="E154">A127</font></i></a>, <a class="txt" href="graphsp.html#E3:98_4_2_2_8_1_1_2"><i><font color="Green" title="E157">A131</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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="E3:98_4_2_2_8"/>
<font color="Maroon" title="c32">Q</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="ref" href="graph_5.html#D19" title="GRAPH_5:def.19">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" title="c23">m</font>, <font color="Maroon" title="c24">j</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>assume </b><a NAME="E1:98_5"/><i><font color="Green" title="E115">A137</font></i>: 
( <font color="Maroon" title="c5">h</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">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="Maroon" title="c24">j</font> &amp; <font color="Maroon" title="c24">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="Maroon" title="c23">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font> )
 ;<br/><b>set </b><font color="Maroon" title="c25">nj</font> = <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font>;<br/><a NAME="E2:98_5"/><i><font color="Green" title="E116">A138</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> &amp; <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98_5"><i><font color="Green" title="E115">A137</font></i></a>, <a class="txt" href="graphsp.html#E83"><i><font color="Green" title="E72">Lm12</font></i></a></i>;<br/><a NAME="E3:98_5"/><b>then </b><i><font color="Green" title="E117">A139</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</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="c4">g</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:98"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="graphsp.html#E15:98"><i><font color="Green" title="E94">A13</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:98_5"/>
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98_5"><i><font color="Green" title="E115">A137</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E5:98_5"/><b>then </b><i><font color="Green" title="E118">A140</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><i><font color="Green" title="E119">A141</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:98_5_1"/>
<span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font>
 ;<br/><a NAME="E2:98_5_1"/><b>then </b>
<font color="Maroon" title="c5">h</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E7:98"><i><font color="Green" title="E88">A7</font></i></a>, <a class="txt" href="graphsp.html#E9:98"><i><font color="Green" title="E89">A8</font></i></a>, <a class="txt" href="graphsp.html#E34:98"><i><font color="Green" title="E108">A27</font></i></a>, <a class="txt" href="graphsp.html#E2:98_5"><i><font color="Green" title="E116">A138</font></i></a>, <a class="txt" href="graphsp.html#E3:98_5"><i><font color="Green" title="E117">A139</font></i></a>, <a class="txt" href="graphsp.html#E5:98_5"><i><font color="Green" title="E118">A140</font></i></a>, <a class="ref" href="graphsp.html#D14" target="_self" title="GRAPHSP:def.14">Def14</a></i>;<br/><b>hence </b><a NAME="E3:98_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98_5"><i><font color="Green" title="E115">A137</font></i></a>, <a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/></div><b>end;</b></div><a NAME="E7:98_5"/><i><font color="Green" title="E120">A142</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E6:98"><i><font color="Green" title="E87">A6</font></i></a>, <a class="txt" href="graphsp.html#E16:98"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="graphsp.html#E2:98_5"><i><font color="Green" title="E116">A138</font></i></a>, <a class="txt" href="graphsp.html#E3:98_5"><i><font color="Green" title="E117">A139</font></i></a>, <a class="txt" href="graphsp.html#E5:98_5"><i><font color="Green" title="E118">A140</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self" title="GRAPHSP:th.19">Th19</a></i>;<br/><a NAME="E8:98_5"/><b>then </b><i><font color="Green" title="E121">A143</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">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
 <b>by </b><i><a class="txt" href="graphsp.html#E7:98"><i><font color="Green" title="E88">A7</font></i></a>, <a class="txt" href="graphsp.html#E9:98"><i><font color="Green" title="E89">A8</font></i></a>, <a class="txt" href="graphsp.html#E1:98_5"><i><font color="Green" title="E115">A137</font></i></a>, <a class="txt" href="graphsp.html#E2:98_5"><i><font color="Green" title="E116">A138</font></i></a>, <a class="txt" href="graphsp.html#E3:98_5"><i><font color="Green" title="E117">A139</font></i></a>, <a class="txt" href="graphsp.html#E5:98_5"><i><font color="Green" title="E118">A140</font></i></a>, <a class="txt" href="graphsp.html#E6:98_5"><i><font color="Green" title="E119">A141</font></i></a>, <a class="ref" href="graphsp.html#D14" target="_self" title="GRAPHSP:def.14">Def14</a></i>;<br/><a NAME="E9:98_5"/><b>then </b><i><font color="Green" title="E122">A144</font></i>: 
not <font color="Maroon" title="c24">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:98"><i><font color="Green" title="E84">A3</font></i></a></i>;<br/><a NAME="E10:98_5"/>
<font color="Maroon" title="c24">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E19:98"><i><font color="Green" title="E98">A17</font></i></a>, <a class="txt" href="graphsp.html#E1:98_5"><i><font color="Green" title="E115">A137</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:98_5"/><b>then </b><i><font color="Green" title="E123">A145</font></i>: 
<font color="Maroon" title="c24">j</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="c4">g</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:98"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="graphsp.html#E15:98"><i><font color="Green" title="E94">A13</font></i></a>, <a class="txt" href="graphsp.html#E1:98_5"><i><font color="Green" title="E115">A137</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:98_5"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c24">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98_5"><i><font color="Green" title="E115">A137</font></i></a>, <a class="txt" href="graphsp.html#E9:98_5"><i><font color="Green" title="E122">A144</font></i></a></i>;<br/><a NAME="E13:98_5"/><b>then </b><i><font color="Green" title="E124">A146</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c24">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E6:98"><i><font color="Green" title="E87">A6</font></i></a>, <a class="txt" href="graphsp.html#E16:98"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="graphsp.html#E19:98"><i><font color="Green" title="E98">A17</font></i></a>, <a class="txt" href="graphsp.html#E1:98_5"><i><font color="Green" title="E115">A137</font></i></a>, <a class="txt" href="graphsp.html#E8:98_5"><i><font color="Green" title="E121">A143</font></i></a>, <a class="txt" href="graphsp.html#E11:98_5"><i><font color="Green" title="E123">A145</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self" title="GRAPHSP:th.19">Th19</a></i>;<br/><a NAME="E14:98_5"/>
not <span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c24">j</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E6:98_5"><i><font color="Green" title="E119">A141</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E15:98_5"/><b>then </b><i><font color="Green" title="E125">A147</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="graphsp.html#E7:98_5"><i><font color="Green" title="E120">A142</font></i></a>, <a class="txt" href="graphsp.html#E8:98_5"><i><font color="Green" title="E121">A143</font></i></a>, <a class="txt" href="graphsp.html#E13:98_5"><i><font color="Green" title="E124">A146</font></i></a>, <a class="ref" href="graphsp.html#D13" target="_self" title="GRAPHSP:def.13">Def13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">v3</font> = <font color="Maroon" title="c24">j</font> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c6">G</font> <b>by </b><i><a class="txt" href="graphsp.html#E10:98"><i><font color="Green" title="E90">A9</font></i></a>, <a class="txt" href="graphsp.html#E1:98_5"><i><font color="Green" title="E115">A137</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>set </b><font color="Maroon" title="c27">Akj</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font>;<br/><a NAME="E17:98_5"/><i><font color="Green" title="E126">A148</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> &amp;  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graphsp.html#E16:98"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="graphsp.html#E1:98_5"><i><font color="Green" title="E115">A137</font></i></a>, <a class="txt" href="graphsp.html#E84"><i><font color="Green" title="E73">Lm13</font></i></a></i>;<br/><a NAME="E18:98_5"/><b>then </b><i><font color="Green" title="E127">A149</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</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="c4">g</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:98"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="graphsp.html#E15:98"><i><font color="Green" title="E94">A13</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E19:98_5"/><i><font color="Green" title="E128">A150</font></i>: 
( <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="graphsp.html#E16:98"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="graphsp.html#E1:98_5"><i><font color="Green" title="E115">A137</font></i></a>, <a class="txt" href="graphsp.html#E85"><i><font color="Green" title="E74">Lm14</font></i></a></i>;<br/><a NAME="E20:98_5"/><i><font color="Green" title="E129">A151</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p5">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</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="c17">vk</font>,<font color="Maroon" title="c26">v3</font>,<font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="ref" href="graphsp.html#D20" target="_self" title="GRAPHSP:def.20">Def20</a></i>;<br/><i><font color="Green" title="E130">A152</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><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="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c24">j</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p5">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E7:98"><i><font color="Green" title="E88">A7</font></i></a>, <a class="txt" href="graphsp.html#E34:98"><i><font color="Green" title="E108">A27</font></i></a>, <a class="txt" href="graphsp.html#E18:98_5"><i><font color="Green" title="E127">A149</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p5">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">j</font></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E6:98"><i><font color="Green" title="E87">A6</font></i></a>, <a class="txt" href="graphsp.html#E17:98_5"><i><font color="Green" title="E126">A148</font></i></a>, <a class="txt" href="graphsp.html#E18:98_5"><i><font color="Green" title="E127">A149</font></i></a>, <a class="ref" href="graphsp.html#T19" target="_self" title="GRAPHSP:th.19">Th19</a></i>
<br/>.= 
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> <font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c26">v3</font>,<font color="Maroon" title="c7">W</font>
<b>by </b><i><a class="txt" href="graphsp.html#E15:98"><i><font color="Green" title="E94">A13</font></i></a>, <a class="txt" href="graphsp.html#E28:98"><i><font color="Green" title="E104">A23</font></i></a>, <a class="txt" href="graphsp.html#E18:98_5"><i><font color="Green" title="E127">A149</font></i></a>, <a class="txt" href="graphsp.html#E19:98_5"><i><font color="Green" title="E128">A150</font></i></a>, <a class="txt" href="graphsp.html#E20:98_5"><i><font color="Green" title="E129">A151</font></i></a>, <a class="ref" href="graphsp.html#D16" target="_self" title="GRAPHSP:def.16">Def16</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_5_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_5_3"/>
( <font color="Maroon" title="c23">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c23">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> )
 </span><b>by </b><i><a class="txt" href="graphsp.html#E17:98"><i><font color="Green" title="E96">A15</font></i></a>, <a class="txt" href="graphsp.html#E1:98_5"><i><font color="Green" title="E115">A137</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_5_3_1"><b>suppose </b></a><a NAME="E1:98_5_3_1"/>
<font color="Maroon" title="c23">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:98_5_3_1"/><b>then </b><i><font color="Green" title="E131">A153</font></i>: 
<font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:98_5_3_1"/>
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">e</font> <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> or  not <font color="Olive" title="b1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c26">v3</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E15:98_5"><i><font color="Green" title="E125">A147</font></i></a>, <a class="txt" href="graphsp.html#E21:98_5"><i><font color="Green" title="E130">A152</font></i></a>, <a class="ref" href="graphsp.html#T24" target="_self" title="GRAPHSP:th.24">Th24</a></i>;<br/><a NAME="E4:98_5_3_1"/><b>then </b>
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> <font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c26">v3</font>,<font color="Maroon" title="c7">W</font> <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="ref" href="graphsp.html#D7" target="_self" title="GRAPHSP:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E5:98_5_3_1"/>
<font color="Maroon" title="c3">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<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="c23">m</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">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
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E2:98_5_3_1"><i><font color="Green" title="E131">A153</font></i></a>, <a class="ref" href="graphsp.html#D20" target="_self" title="GRAPHSP:def.20">Def20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_5_3_2"><b>suppose </b></a><a NAME="E1:98_5_3_2"/>
<font color="Maroon" title="c23">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:98_5_3_2"/>
<font color="Maroon" title="c3">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</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" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<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="c23">m</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">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
 <b>by </b><i><a class="txt" href="graphsp.html#E3:98"><i><font color="Green" title="E84">A3</font></i></a>, <a class="txt" href="graphsp.html#E1:98_5"><i><font color="Green" title="E115">A137</font></i></a>, <a class="txt" href="graphsp.html#E8:98_5"><i><font color="Green" title="E121">A143</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" title="c23">m</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>assume </b><a NAME="E46:98"/><i><font color="Green" title="E115">A154</font></i>: 
<font color="Maroon" title="c23">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_6"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_6"/>
( <font color="Maroon" title="c23">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c23">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 </span><b>by </b><i><a class="txt" href="graphsp.html#E17:98"><i><font color="Green" title="E96">A15</font></i></a>, <a class="txt" href="graphsp.html#E46:98"><i><font color="Green" title="E115">A154</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_6_1"><b>suppose </b></a><a NAME="E1:98_6_1"/><i><font color="Green" title="E116">A155</font></i>: 
<font color="Maroon" title="c23">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><a NAME="E2:98_6_1"/><b>then </b>
<font color="Maroon" title="c5">h</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:98"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="graphsp.html#E5:98"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="graphsp.html#E89"><i><font color="Green" title="E77">Lm16</font></i></a></i>;<br/><b>hence </b><a NAME="E3:98_6_1"/>
<font color="Maroon" title="c5">h</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E3:98"><i><font color="Green" title="E84">A3</font></i></a>, <a class="txt" href="graphsp.html#E1:98_6_1"><i><font color="Green" title="E116">A155</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_6_2"><b>suppose </b></a><a NAME="E1:98_6_2"/>
<font color="Maroon" title="c23">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:98_6_2"/>
<font color="Maroon" title="c5">h</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E16:98"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="graphsp.html#E43:98"><i><font color="Green" title="E114">A34</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
