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<span class="kw">let </span><font color="Maroon" title="c1">e</font>, <font color="Maroon" title="c2">V</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">W</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/> for <font color="Olive" title="b2">G</font> being  <a href="graph_1.html#V2" title="GRAPH_1:attr.2">oriented</a>   <a href="graph_1.html#V6" title="GRAPH_1:attr.6">finite</a>  <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a><br/> for <font color="Olive" title="b3">P</font>, <font color="Olive" title="b4">Q</font>, <font color="Olive" title="b5">R</font> being  <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Olive" title="b2">G</font><br/> for <font color="Olive" title="b6">v1</font>, <font color="Olive" title="b7">v2</font>, <font color="Olive" title="b8">v3</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">G</font>  st <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b1">W</font> <a href="graph_5.html#R5" title="GRAPH_5:pred.5">is_weight&gt;=0of</a> <font color="Olive" title="b2">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">P</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b3">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b6">v1</font>,<font color="Olive" title="b7">v2</font>,<font color="Maroon" title="c2">V</font>,<font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b6">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">v2</font> &amp; <font color="Olive" title="b6">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b8">v3</font> &amp; <font color="Olive" title="b5">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">P</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">e</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b4">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b6">v1</font>,<font color="Olive" title="b8">v3</font>,<font color="Maroon" title="c2">V</font>,<font color="Olive" title="b1">W</font> &amp; <font color="Maroon" title="c1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Olive" title="b7">v2</font>,<font color="Olive" title="b8">v3</font> &amp; <font color="Olive" title="b3">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a> <font color="Maroon" title="c2">V</font>,<font color="Olive" title="b6">v1</font>,<font color="Olive" title="b1">W</font> holds <br/>( (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b4">Q</font>,<font color="Olive" title="b1">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="Olive" title="b5">R</font>,<font color="Olive" title="b1">W</font> implies <font color="Olive" title="b4">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b6">v1</font>,<font color="Olive" title="b8">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Olive" title="b1">W</font> ) &amp; (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b4">Q</font>,<font color="Olive" title="b1">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b5">R</font>,<font color="Olive" title="b1">W</font> implies <font color="Olive" title="b5">R</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b6">v1</font>,<font color="Olive" title="b8">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Olive" title="b1">W</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">W</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <a href="graph_1.html#V2" title="GRAPH_1:attr.2">oriented</a>   <a href="graph_1.html#V6" title="GRAPH_1:attr.6">finite</a>  <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a><br/> for <font color="Olive" title="b2">P</font>, <font color="Olive" title="b3">Q</font>, <font color="Olive" title="b4">R</font> being  <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Olive" title="b1">G</font><br/> for <font color="Olive" title="b5">v1</font>, <font color="Olive" title="b6">v2</font>, <font color="Olive" title="b7">v3</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">G</font>  st <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c3">W</font> <a href="graph_5.html#R5" title="GRAPH_5:pred.5">is_weight&gt;=0of</a> <font color="Olive" title="b1">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">P</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b2">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b5">v1</font>,<font color="Olive" title="b6">v2</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font> &amp; <font color="Olive" title="b5">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">v2</font> &amp; <font color="Olive" title="b5">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">v3</font> &amp; <font color="Olive" title="b4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">e</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b3">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b5">v1</font>,<font color="Olive" title="b7">v3</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font> &amp; <font color="Maroon" title="c1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Olive" title="b6">v2</font>,<font color="Olive" title="b7">v3</font> &amp; <font color="Olive" title="b2">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a> <font color="Maroon" title="c2">V</font>,<font color="Olive" title="b5">v1</font>,<font color="Maroon" title="c3">W</font> holds <br/>( (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b3">Q</font>,<font color="Maroon" title="c3">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="Olive" title="b4">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Olive" title="b3">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b5">v1</font>,<font color="Olive" title="b7">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b6">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) &amp; (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b3">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b4">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Olive" title="b4">R</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b5">v1</font>,<font color="Olive" title="b7">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b6">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">G</font> be   <a href="graph_1.html#V2" title="GRAPH_1:attr.2">oriented</a>   <a href="graph_1.html#V6" title="GRAPH_1:attr.6">finite</a>  <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font>, <font color="Olive" title="b3">R</font> being  <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c4">G</font><br/> for <font color="Olive" title="b4">v1</font>, <font color="Olive" title="b5">v2</font>, <font color="Olive" title="b6">v3</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font>  st <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c3">W</font> <a href="graph_5.html#R5" title="GRAPH_5:pred.5">is_weight&gt;=0of</a> <font color="Maroon" title="c4">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">P</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b4">v1</font>,<font color="Olive" title="b5">v2</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font> &amp; <font color="Olive" title="b4">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">v2</font> &amp; <font color="Olive" title="b4">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">v3</font> &amp; <font color="Olive" title="b3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">e</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b2">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b4">v1</font>,<font color="Olive" title="b6">v3</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font> &amp; <font color="Maroon" title="c1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Olive" title="b5">v2</font>,<font color="Olive" title="b6">v3</font> &amp; <font color="Olive" title="b1">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a> <font color="Maroon" title="c2">V</font>,<font color="Olive" title="b4">v1</font>,<font color="Maroon" title="c3">W</font> holds <br/>( (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b2">Q</font>,<font color="Maroon" title="c3">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="Olive" title="b3">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Olive" title="b2">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b4">v1</font>,<font color="Olive" title="b6">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b5">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) &amp; (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b2">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b3">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Olive" title="b3">R</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b4">v1</font>,<font color="Olive" title="b6">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b5">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">P</font>, <font color="Maroon" title="c6">Q</font>, <font color="Maroon" title="c7">R</font> be   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c4">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font>, <font color="Olive" title="b3">v3</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font>  st <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c3">W</font> <a href="graph_5.html#R5" title="GRAPH_5:pred.5">is_weight&gt;=0of</a> <font color="Maroon" title="c4">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c5">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font> &amp; <font color="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> &amp; <font color="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">v3</font> &amp; <font color="Maroon" title="c7">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">e</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Maroon" title="c6">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v3</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font> &amp; <font color="Maroon" title="c1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Olive" title="b2">v2</font>,<font color="Olive" title="b3">v3</font> &amp; <font color="Maroon" title="c5">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a> <font color="Maroon" title="c2">V</font>,<font color="Olive" title="b1">v1</font>,<font color="Maroon" title="c3">W</font> holds <br/>( (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c6">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) &amp; (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c7">R</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">v1</font>, <font color="Maroon" title="c9">v2</font>, <font color="Maroon" title="c10">v3</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c3">W</font> <a href="graph_5.html#R5" title="GRAPH_5:pred.5">is_weight&gt;=0of</a> <font color="Maroon" title="c4">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c5">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="c9">v2</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font> &amp; <font color="Maroon" title="c8">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">v2</font> &amp; <font color="Maroon" title="c8">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">v3</font> &amp; <font color="Maroon" title="c7">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">e</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font> &amp; <font color="Maroon" title="c1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c9">v2</font>,<font color="Maroon" title="c10">v3</font> &amp; <font color="Maroon" title="c5">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c3">W</font> implies ( (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) &amp; (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) ) )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:99"/><span class="lab"><font color="Green" title="E71">A1</font></span>: 
<font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c4">G</font>
 <span class="kw">and </span><br/><a NAME="E2:99"/><span class="lab"><font color="Green" title="E72">A2</font></span>: 
<font color="Maroon" title="c3">W</font> <a href="graph_5.html#R5" title="GRAPH_5:pred.5">is_weight&gt;=0of</a> <font color="Maroon" title="c4">G</font>
 <span class="kw">and </span><br/><a NAME="E3:99"/><span class="lab"><font color="Green" title="E73">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E4:99"/><span class="lab"><font color="Green" title="E74">A4</font></span>: 
<font color="Maroon" title="c5">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="c9">v2</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">and </span><br/><a NAME="E5:99"/><span class="lab"><font color="Green" title="E75">A5</font></span>: 
<font color="Maroon" title="c8">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">v2</font>
 <span class="kw">and </span><br/><a NAME="E6:99"/><span class="lab"><font color="Green" title="E76">A6</font></span>: 
<font color="Maroon" title="c8">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">v3</font>
 <span class="kw">and </span><br/><a NAME="E7:99"/><span class="lab"><font color="Green" title="E77">A7</font></span>: 
<font color="Maroon" title="c7">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">e</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E8:99"/><span class="lab"><font color="Green" title="E78">A8</font></span>: 
<font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">and </span><br/><a NAME="E9:99"/><span class="lab"><font color="Green" title="E79">A9</font></span>: 
<font color="Maroon" title="c1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c9">v2</font>,<font color="Maroon" title="c10">v3</font>
 <span class="kw">and </span><br/><a NAME="E10:99"/><span class="lab"><font color="Green" title="E80">A10</font></span>: 
<font color="Maroon" title="c5">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c3">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) &amp; (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) )</span><br/>

<a NAME="E11:99"/><span class="lab"><font color="Green" title="E81">A11</font></span>: 
<font color="Maroon" title="c5">P</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="c9">v2</font>,<font color="Maroon" title="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>;<br/>
<a NAME="E12:99"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A12</font></span>: 
<font color="Maroon" title="c8">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:99"><span class="lab"><font color="Green" title="E75">A5</font></span></a>, <a class="ref" href="graph_5.html#T35" target="_self" title="GRAPH_5:th.35">Th35</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">Eg</font> = the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c4">G</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">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="c1">e</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c4">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="ref" href="finseq_1.html#T95" title="FINSEQ_1:th.95">FINSEQ_1:95</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">V9</font> = <font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E14:99"/>
<font color="Maroon" title="c6">Q</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="c10">v3</font>,<font color="Maroon" title="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:99"><span class="lab"><font color="Green" title="E78">A8</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>;<br/>
<a NAME="E15:99"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A13</font></span>: 
<font color="Maroon" title="c6">Q</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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#T36" target="_self" title="GRAPH_5:th.36">Th36</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E16:99"/><span class="lab"><font color="Green" title="E84">A14</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">pe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c12">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="c1">e</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">s</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c4">G</font><span class="kw"> such that </span><br/><a NAME="E18:99"/><span class="lab"><font color="Green" title="E85">A15</font></span>: 
<font color="Maroon" title="c14">s</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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E73">A3</font></span></a>, <a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E77">A7</font></span></a>, <a class="txt" href="#E9:99"><span class="lab"><font color="Green" title="E79">A9</font></span></a>, <a class="txt" href="#E11:99"><span class="lab"><font color="Green" title="E81">A11</font></span></a>, <a class="ref" href="graph_5.html#T38" target="_self" title="GRAPH_5:th.38">Th38</a>, <a class="ref" href="graph_5.html#T66" target="_self" title="GRAPH_5:th.66">Th66</a></span>;<br/>
<a NAME="E19:99"/><span class="lab"><font color="Green" title="E86">A16</font></span>: 
<font color="Maroon" title="c7">R</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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E73">A3</font></span></a>, <a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E77">A7</font></span></a>, <a class="txt" href="#E9:99"><span class="lab"><font color="Green" title="E79">A9</font></span></a>, <a class="txt" href="#E11:99"><span class="lab"><font color="Green" title="E81">A11</font></span></a>, <a class="txt" href="#E16:99"><span class="lab"><font color="Green" title="E84">A14</font></span></a>, <a class="ref" href="graph_5.html#T38" target="_self" title="GRAPH_5:th.38">Th38</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E87">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:99_1"/><span class="lab"><font color="Green" title="E88">A18</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c14">s</font>,<font color="Maroon" title="c3">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) &amp; (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:99_1_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c7">R</font>,<font color="Maroon" title="c3">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">u</font> be   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c4">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">u</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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c15">u</font>,<font color="Maroon" title="c3">W</font> )</span><br/><span class="kw">assume </span><a NAME="E1:99_1_1_1"/>
<font color="Maroon" title="c15">u</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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c15">u</font>,<font color="Maroon" title="c3">W</font></span><br/><a NAME="E2:99_1_1_1"/><span class="kw">then </span>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c3">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="c15">u</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:99"><span class="lab"><font color="Green" title="E85">A15</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>;<br/><span class="kw">hence </span><a NAME="E3:99_1_1_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c15">u</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_1"><span class="lab"><font color="Green" title="E88">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:99_1_1"/>
<font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:99"><span class="lab"><font color="Green" title="E83">A13</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:99_1_2"/><span class="lab"><font color="Green" title="E89">A19</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">u</font> be   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c4">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">u</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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">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="c15">u</font>,<font color="Maroon" title="c3">W</font> )</span><br/><span class="kw">assume </span><a NAME="E1:99_1_2_1"/>
<font color="Maroon" title="c15">u</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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">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="c15">u</font>,<font color="Maroon" title="c3">W</font></span><br/><a NAME="E2:99_1_2_1"/><span class="kw">then </span>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c3">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="c15">u</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:99"><span class="lab"><font color="Green" title="E85">A15</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>;<br/><a NAME="E3:99_1_2_1"/><span class="kw">then </span>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c15">u</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_1"><span class="lab"><font color="Green" title="E88">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:99_1_2_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">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="c15">u</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_1_2"><span class="lab"><font color="Green" title="E89">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:99_1_2"/>
<font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:99"><span class="lab"><font color="Green" title="E86">A16</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c15">FT</font> = the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c4">G</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">FS</font> = the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">G</font>;<br/>
<a NAME="E21:99"/><span class="lab"><font color="Green" title="E88">A20</font></span>: 
( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v2</font> &amp; the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99"><span class="lab"><font color="Green" title="E79">A9</font></span></a>, <a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></span>;<br/>
<a NAME="E22:99"/><span class="lab"><font color="Green" title="E89">A21</font></span>: 
<font color="Maroon" title="c14">s</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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:99"><span class="lab"><font color="Green" title="E85">A15</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>;<br/>
<a NAME="E23:99"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A22</font></span>: 
<font color="Maroon" title="c14">s</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="c10">v3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/>
<a NAME="E24:99"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A23</font></span>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">s</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/>
<a NAME="E25:99"/>
<font color="Maroon" title="c14">s</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:99"><span class="lab"><font color="Green" title="E90">A22</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/>
<a NAME="E26:99"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A24</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">s</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:99_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">s</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_1"><span class="kw">suppose </span></a><a NAME="E1:99_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">s</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) &amp; (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:99_2_1"/><span class="lab"><font color="Green" title="E93">A25</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><a NAME="E4:99_2_1"/><span class="lab"><font color="Green" title="E94">A26</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T40" title="XBOOLE_1:th.40">XBOOLE_1:40</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">k</font> = <font color="Maroon" title="c17">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E6:99_2_1"/><span class="lab"><font color="Green" title="E95">A27</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_2_1"><span class="lab"><font color="Green" title="E93">A25</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">s1</font>, <font color="Maroon" title="c20">s2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E8:99_2_1"/><span class="lab"><font color="Green" title="E96">A28</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k</font>
 <span class="kw">and </span><br/><a NAME="E9:99_2_1"/><span class="lab"><font color="Green" title="E97">A29</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E10:99_2_1"/><span class="lab"><font color="Green" title="E98">A30</font></span>: 
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">s1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c20">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T25" title="FINSEQ_2:th.25">FINSEQ_2:25</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">s1</font> = <font color="Maroon" title="c19">s1</font>, <font color="Maroon" title="c22">s2</font> = <font color="Maroon" title="c20">s2</font> as   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c4">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:99_2_1"><span class="lab"><font color="Green" title="E98">A30</font></span></a>, <a class="ref" href="graph_5.html#T17" target="_self" title="GRAPH_5:th.17">Th17</a></span>;<br/><a NAME="E12:99_2_1"/><span class="lab"><font color="Green" title="E99">A31</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">s1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:99_2_1"><span class="lab"><font color="Green" title="E96">A28</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E13:99_2_1"/><span class="lab"><font color="Green" title="E100">A32</font></span>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:99"><span class="lab"><font color="Green" title="E90">A22</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E14:99_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A33</font></span>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:99_2_1"><span class="lab"><font color="Green" title="E98">A30</font></span></a>, <a class="txt" href="#E12:99_2_1"><span class="lab"><font color="Green" title="E99">A31</font></span></a>, <a class="txt" href="graph_5.html#E8"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/><a NAME="E15:99_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_2_1"><span class="lab"><font color="Green" title="E97">A29</font></span></a></span>;<br/><a NAME="E16:99_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A34</font></span>: 
not <font color="Maroon" title="c10">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c21">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99"><span class="lab"><font color="Green" title="E76">A6</font></span></a>, <a class="txt" href="#E24:99"><span class="lab"><font color="Green" title="E91">A23</font></span></a>, <a class="txt" href="#E10:99_2_1"><span class="lab"><font color="Green" title="E98">A30</font></span></a>, <a class="txt" href="#E12:99_2_1"><span class="lab"><font color="Green" title="E99">A31</font></span></a>, <a class="txt" href="#E13:99_2_1"><span class="lab"><font color="Green" title="E100">A32</font></span></a>, <a class="ref" href="graph_5.html#T21" target="_self" title="GRAPH_5:th.21">Th21</a></span>;<br/><a NAME="E17:99_2_1"/><span class="lab"><font color="Green" title="E103">A35</font></span>: 
<font color="Maroon" title="c22">s2</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="c14">s</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="c14">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99_2_1"><span class="lab"><font color="Green" title="E95">A27</font></span></a>, <a class="txt" href="#E8:99_2_1"><span class="lab"><font color="Green" title="E96">A28</font></span></a>, <a class="txt" href="#E9:99_2_1"><span class="lab"><font color="Green" title="E97">A29</font></span></a>, <a class="txt" href="#E10:99_2_1"><span class="lab"><font color="Green" title="E98">A30</font></span></a>, <a class="txt" href="graph_5.html#E9"><span class="lab"><font color="Green" title="E7">Lm2</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A36</font></span>: <span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c14">s</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c21">s1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:99"><span class="lab"><font color="Green" title="E91">A23</font></span></a>, <a class="txt" href="#E8:99_2_1"><span class="lab"><font color="Green" title="E96">A28</font></span></a>, <a class="txt" href="#E9:99_2_1"><span class="lab"><font color="Green" title="E97">A29</font></span></a>, <a class="txt" href="#E10:99_2_1"><span class="lab"><font color="Green" title="E98">A30</font></span></a>, <a class="ref" href="graph_5.html#T31" target="_self" title="GRAPH_5:th.31">Th31</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c21">s1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T40" title="XBOOLE_1:th.40">XBOOLE_1:40</a></span>
<br/>.= 
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c21">s1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:99_2_1"><span class="lab"><font color="Green" title="E102">A34</font></span></a>, <a class="ref" href="zfmisc_1.html#T65" title="ZFMISC_1:th.65">ZFMISC_1:65</a></span>
;<br/><a NAME="E19:99_2_1"/><span class="kw">then </span>
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c21">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:99"><span class="lab"><font color="Green" title="E89">A21</font></span></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/><a NAME="E20:99_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c21">s1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T33" title="XBOOLE_1:th.33">XBOOLE_1:33</a></span>;<br/><a NAME="E21:99_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A37</font></span>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c21">s1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_2_1"><span class="lab"><font color="Green" title="E94">A26</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E22:99_2_1"/><span class="lab"><font color="Green" title="E106">A38</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">s1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99_2_1"><span class="lab"><font color="Green" title="E95">A27</font></span></a>, <a class="txt" href="#E8:99_2_1"><span class="lab"><font color="Green" title="E96">A28</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E23:99_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A39</font></span>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">s1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">s1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:99_2_1"><span class="lab"><font color="Green" title="E99">A31</font></span></a>, <a class="ref" href="graph_1.html#D13" title="GRAPH_1:def.13">GRAPH_1:def 13</a></span>;<br/><a NAME="E24:99_2_1"/><span class="lab"><font color="Green" title="E108">A40</font></span>: 
<font color="Maroon" title="c21">s1</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="c21">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</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="c21">s1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:99_2_1"><span class="lab"><font color="Green" title="E98">A30</font></span></a>, <a class="txt" href="#E12:99_2_1"><span class="lab"><font color="Green" title="E99">A31</font></span></a>, <a class="txt" href="graph_5.html#E8"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/><a NAME="E25:99_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A41</font></span>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">s1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99_2_1"><span class="lab"><font color="Green" title="E95">A27</font></span></a>, <a class="txt" href="#E8:99_2_1"><span class="lab"><font color="Green" title="E96">A28</font></span></a>, <a class="txt" href="#E17:99_2_1"><span class="lab"><font color="Green" title="E103">A35</font></span></a>, <a class="txt" href="#E12:99_2_1"><span class="lab"><font color="Green" title="E99">A31</font></span></a>, <a class="txt" href="#E22:99_2_1"><span class="lab"><font color="Green" title="E106">A38</font></span></a>, <a class="ref" href="graph_1.html#D13" title="GRAPH_1:def.13">GRAPH_1:def 13</a></span>;<br/><a NAME="E26:99_2_1"/><span class="lab"><font color="Green" title="E110">A42</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c21">s1</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c22">s2</font>,<font color="Maroon" title="c3">W</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="txt" href="#E10:99_2_1"><span class="lab"><font color="Green" title="E98">A30</font></span></a>, <a class="ref" href="graph_5.html#T50" target="_self" title="GRAPH_5:th.50">Th50</a>, <a class="ref" href="graph_5.html#T58" target="_self" title="GRAPH_5:th.58">Th58</a></span>;<br/><a NAME="E27:99_2_1"/><span class="lab"><font color="Green" title="E111">A43</font></span>: 
not <font color="Maroon" title="c8">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c22">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99"><span class="lab"><font color="Green" title="E76">A6</font></span></a>, <a class="txt" href="#E24:99"><span class="lab"><font color="Green" title="E91">A23</font></span></a>, <a class="txt" href="#E9:99_2_1"><span class="lab"><font color="Green" title="E97">A29</font></span></a>, <a class="txt" href="#E10:99_2_1"><span class="lab"><font color="Green" title="E98">A30</font></span></a>, <a class="txt" href="#E12:99_2_1"><span class="lab"><font color="Green" title="E99">A31</font></span></a>, <a class="txt" href="#E13:99_2_1"><span class="lab"><font color="Green" title="E100">A32</font></span></a>, <a class="ref" href="graph_5.html#T21" target="_self" title="GRAPH_5:th.21">Th21</a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:99_2_1_2_1"/>
( <font color="Maroon" title="c9">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c21">s1</font> or  not <font color="Maroon" title="c9">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c21">s1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:99_2_1_2_1_1"/>
<font color="Maroon" title="c9">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c21">s1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) &amp; (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:99_2_1_2_1_1"/><span class="lab"><font color="Green" title="E112">A44</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font>
 <span class="kw">and </span><br/><a NAME="E4:99_2_1_2_1_1"/><span class="lab"><font color="Green" title="E113">A45</font></span>: 
<font color="Maroon" title="c23">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="c21">s1</font>
 <span class="kw">and </span><br/><a NAME="E5:99_2_1_2_1_1"/><span class="lab"><font color="Green" title="E114">A46</font></span>: 
<font color="Maroon" title="c9">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:99"><span class="lab"><font color="Green" title="E75">A5</font></span></a>, <a class="txt" href="#E14:99_2_1"><span class="lab"><font color="Green" title="E101">A33</font></span></a>, <a class="ref" href="graph_5.html#T32" target="_self" title="GRAPH_5:th.32">Th32</a></span>;<br/><a NAME="E6:99_2_1_2_1_1"/>
<font color="Maroon" title="c22">s2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c4">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E71">A1</font></span></a></span>;<br/><a NAME="E7:99_2_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A47</font></span>: 
<font color="Maroon" title="c22">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c4">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_2_1"><span class="lab"><font color="Green" title="E97">A29</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_1_2_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:99_2_1_2_1_1_1_1"/>
( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v2</font> or the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">v2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_1_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:99_2_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E116">A48</font></span>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) &amp; (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) )</span><br/><div class="add"><a NAME="E2:99_2_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c21">s1</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:99_2_1"><span class="lab"><font color="Green" title="E96">A28</font></span></a></span>;<br/><a NAME="E3:99_2_1_2_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">s1</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="c9">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:99_2_1"><span class="lab"><font color="Green" title="E101">A33</font></span></a>, <a class="txt" href="#E25:99_2_1"><span class="lab"><font color="Green" title="E109">A41</font></span></a>, <a class="txt" href="#E1:99_2_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E116">A48</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E4:99_2_1_2_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">s1</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="c9">v2</font>,<font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:99_2_1"><span class="lab"><font color="Green" title="E105">A37</font></span></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/><a NAME="E5:99_2_1_2_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A49</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c3">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="c21">s1</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>;<br/><a NAME="E6:99_2_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c22">s2</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="c1">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="txt" href="#E24:99"><span class="lab"><font color="Green" title="E91">A23</font></span></a>, <a class="txt" href="#E21:99"><span class="lab"><font color="Green" title="E88">A20</font></span></a>, <a class="txt" href="#E17:99_2_1"><span class="lab"><font color="Green" title="E103">A35</font></span></a>, <a class="txt" href="#E7:99_2_1_2_1_1"><span class="lab"><font color="Green" title="E115">A47</font></span></a>, <a class="txt" href="#E1:99_2_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E116">A48</font></span></a>, <a class="ref" href="graph_1.html#D5" title="GRAPH_1:def.5">GRAPH_1:def 5</a></span>;<br/><a NAME="E7:99_2_1_2_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">pe</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_2_1"><span class="lab"><font color="Green" title="E97">A29</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/><a NAME="E8:99_2_1_2_1_1_1_1_1"/><span class="kw">then </span>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c22">s2</font>,<font color="Maroon" title="c3">W</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E77">A7</font></span></a>, <a class="ref" href="graph_5.html#T50" target="_self" title="GRAPH_5:th.50">Th50</a>, <a class="ref" href="graph_5.html#T58" target="_self" title="GRAPH_5:th.58">Th58</a></span>;<br/><a NAME="E9:99_2_1_2_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A50</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">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="c14">s</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:99_2_1"><span class="lab"><font color="Green" title="E110">A42</font></span></a>, <a class="txt" href="#E5:99_2_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E117">A49</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:99_2_1_2_1_1_1_1_1_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c7">R</font>,<font color="Maroon" title="c3">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font></span><br/><a NAME="E2:99_2_1_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A51</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c14">s</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_2_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E118">A50</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_1_2_1_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">u</font> be   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c4">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">u</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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c24">u</font>,<font color="Maroon" title="c3">W</font> )</span><br/><span class="kw">assume </span><a NAME="E1:99_2_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c24">u</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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c24">u</font>,<font color="Maroon" title="c3">W</font></span><br/><a NAME="E2:99_2_1_2_1_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c3">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="c24">u</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:99"><span class="lab"><font color="Green" title="E85">A15</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>;<br/><span class="kw">hence </span><a NAME="E3:99_2_1_2_1_1_1_1_1_1_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c24">u</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_2_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E119">A51</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:99_2_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:99"><span class="lab"><font color="Green" title="E83">A13</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:99_2_1_2_1_1_1_1_1_2"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_1_2_1_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">u</font> be   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c4">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">u</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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">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="c24">u</font>,<font color="Maroon" title="c3">W</font> )</span><br/><span class="kw">assume </span><a NAME="E1:99_2_1_2_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c24">u</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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">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="c24">u</font>,<font color="Maroon" title="c3">W</font></span><br/><a NAME="E2:99_2_1_2_1_1_1_1_1_2_1"/><span class="kw">then </span>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c3">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="c24">u</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:99"><span class="lab"><font color="Green" title="E85">A15</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>;<br/><span class="kw">hence </span><a NAME="E3:99_2_1_2_1_1_1_1_1_2_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">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="c24">u</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_2_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E118">A50</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:99_2_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:99"><span class="lab"><font color="Green" title="E86">A16</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_1_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:99_2_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E116">A52</font></span>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) &amp; (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) )</span><br/><div class="add"><a NAME="E2:99_2_1_2_1_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">s1</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="c21">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:99_2_1"><span class="lab"><font color="Green" title="E99">A31</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">vx</font> = the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">s1</font></span>)</span></span>)</span> as   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c4">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><a NAME="E4:99_2_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E117">A53</font></span>: 
not <font color="Maroon" title="c24">vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:99_2_1"><span class="lab"><font color="Green" title="E109">A41</font></span></a>, <a class="txt" href="#E1:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E116">A52</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E5:99_2_1_2_1_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">s1</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="c21">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:99_2_1"><span class="lab"><font color="Green" title="E99">A31</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E6:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A54</font></span>: 
<font color="Maroon" title="c24">vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c21">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#T28" target="_self" title="GRAPH_5:th.28">Th28</a></span>;<br/><a NAME="E7:99_2_1_2_1_1_1_1_2"/>
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c21">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:99"><span class="lab"><font color="Green" title="E89">A21</font></span></a>, <a class="txt" href="#E18:99_2_1"><span class="lab"><font color="Green" title="E104">A36</font></span></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/><a NAME="E8:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A55</font></span>: 
<font color="Maroon" title="c24">vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E118">A54</font></span></a>, <a class="txt" href="#E4:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E117">A53</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E9:99_2_1_2_1_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c24">vx</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_1_2_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c24">vx</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:99_2_1_2_1_1_1_1_2_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c24">vx</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font></span><br/>

<span class="kw">hence </span><a NAME="E2:99_2_1_2_1_1_1_1_2_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E119">A55</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E10:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A56</font></span>: 
<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c24">vx</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><a NAME="E11:99_2_1_2_1_1_1_1_2"/>
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="c22">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_2_1"><span class="lab"><font color="Green" title="E97">A29</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E12:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">vx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_2_1"><span class="lab"><font color="Green" title="E93">A25</font></span></a>, <a class="txt" href="#E8:99_2_1"><span class="lab"><font color="Green" title="E96">A28</font></span></a>, <a class="txt" href="#E17:99_2_1"><span class="lab"><font color="Green" title="E103">A35</font></span></a>, <a class="txt" href="#E24:99_2_1"><span class="lab"><font color="Green" title="E108">A40</font></span></a>, <a class="txt" href="#E27:99_2_1"><span class="lab"><font color="Green" title="E111">A43</font></span></a>, <a class="txt" href="#E23:99_2_1"><span class="lab"><font color="Green" title="E107">A39</font></span></a>, <a class="ref" href="graph_5.html#T28" target="_self" title="GRAPH_5:th.28">Th28</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">q9</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c4">G</font><span class="kw"> such that </span><br/><a NAME="E14:99_2_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E121">A57</font></span>: 
<font color="Maroon" title="c25">q9</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="c24">vx</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">and </span><br/><a NAME="E15:99_2_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E122">A58</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c25">q9</font>,<font color="Maroon" title="c3">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="c5">P</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:99"><span class="lab"><font color="Green" title="E80">A10</font></span></a>, <a class="txt" href="#E8:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E119">A55</font></span></a>, <a class="ref" href="graph_5.html#D19" target="_self" title="GRAPH_5:def.19">Def19</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c26">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E17:99_2_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E123">A59</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_2_1_2_1_1"><span class="lab"><font color="Green" title="E113">A45</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><a NAME="E18:99_2_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E124">A60</font></span>: 
<font color="Maroon" title="c25">q9</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="c24">vx</font>,<font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E121">A57</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>;<br/><a NAME="E19:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A61</font></span>: 
<font color="Maroon" title="c25">q9</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="c24">vx</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/><a NAME="E20:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c25">q9</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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E21:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A62</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">q9</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></span>;<br/><a NAME="E22:99_2_1_2_1_1_1_1_2"/>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">q9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">q9</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">vx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E125">A61</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">qx</font> = <font color="Maroon" title="c25">q9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c22">s2</font> as   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c4">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_2_1"><span class="lab"><font color="Green" title="E93">A25</font></span></a>, <a class="txt" href="#E8:99_2_1"><span class="lab"><font color="Green" title="E96">A28</font></span></a>, <a class="txt" href="#E17:99_2_1"><span class="lab"><font color="Green" title="E103">A35</font></span></a>, <a class="txt" href="#E24:99_2_1"><span class="lab"><font color="Green" title="E108">A40</font></span></a>, <a class="txt" href="#E23:99_2_1"><span class="lab"><font color="Green" title="E107">A39</font></span></a>, <a class="ref" href="graph_5.html#T15" target="_self" title="GRAPH_5:th.15">Th15</a></span>;<br/><a NAME="E24:99_2_1_2_1_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">qx</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="c25">q9</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_2_1"><span class="lab"><font color="Green" title="E97">A29</font></span></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/><a NAME="E25:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A63</font></span>: 
( <font color="Maroon" title="c27">qx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">qx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">qx</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:99"><span class="lab"><font color="Green" title="E91">A23</font></span></a>, <a class="txt" href="#E9:99_2_1"><span class="lab"><font color="Green" title="E97">A29</font></span></a>, <a class="txt" href="#E17:99_2_1"><span class="lab"><font color="Green" title="E103">A35</font></span></a>, <a class="txt" href="graph_5.html#E9"><span class="lab"><font color="Green" title="E7">Lm2</font></span></a></span>;<br/><a NAME="E26:99_2_1_2_1_1_1_1_2"/>
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">q9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E125">A61</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E27:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span>
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">qx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E126">A62</font></span></a>, <a class="txt" href="graph_5.html#E8"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/><a NAME="E28:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A64</font></span>: 
<font color="Maroon" title="c27">qx</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="c10">v3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E127">A63</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E29:99_2_1_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c25">q9</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c24">vx</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E124">A60</font></span></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/><a NAME="E30:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span>
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c25">q9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c24">vx</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T44" title="XBOOLE_1:th.44">XBOOLE_1:44</a></span>;<br/><a NAME="E31:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span>
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c25">q9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E120">A56</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E32:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A65</font></span>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c25">q9</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T33" title="XBOOLE_1:th.33">XBOOLE_1:33</a></span>;<br/><a NAME="E33:99_2_1_2_1_1_1_1_2"/>
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c27">qx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c25">q9</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:99"><span class="lab"><font color="Green" title="E91">A23</font></span></a>, <a class="txt" href="#E9:99_2_1"><span class="lab"><font color="Green" title="E97">A29</font></span></a>, <a class="txt" href="#E17:99_2_1"><span class="lab"><font color="Green" title="E103">A35</font></span></a>, <a class="txt" href="#E21:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E126">A62</font></span></a>, <a class="ref" href="graph_5.html#T31" target="_self" title="GRAPH_5:th.31">Th31</a></span>;<br/><a NAME="E34:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c27">qx</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c25">q9</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T40" title="XBOOLE_1:th.40">XBOOLE_1:40</a></span>;<br/><a NAME="E35:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c27">qx</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E129">A65</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E36:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c27">qx</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="c10">v3</font>,<font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E128">A64</font></span></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/><a NAME="E37:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A66</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c27">qx</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:99"><span class="lab"><font color="Green" title="E78">A8</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c28">j</font> = <font color="Maroon" title="c26">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c29">t1</font>, <font color="Maroon" title="c30">t2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E40:99_2_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E131">A67</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i</font>
 <span class="kw">and </span><br/><a NAME="E41:99_2_1_2_1_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">j</font>
 <span class="kw">and </span><br/><a NAME="E42:99_2_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E132">A68</font></span>: 
<font color="Maroon" title="c21">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">t1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c30">t2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E123">A59</font></span></a>, <a class="ref" href="finseq_2.html#T25" title="FINSEQ_2:th.25">FINSEQ_2:25</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c31">t1</font> = <font color="Maroon" title="c29">t1</font>, <font color="Maroon" title="c32">t2</font> = <font color="Maroon" title="c30">t2</font> as   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c4">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E132">A68</font></span></a>, <a class="ref" href="graph_5.html#T17" target="_self" title="GRAPH_5:th.17">Th17</a></span>;<br/><a NAME="E44:99_2_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E133">A69</font></span>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c31">t1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_2_1_2_1_1"><span class="lab"><font color="Green" title="E112">A44</font></span></a>, <a class="txt" href="#E5:99_2_1_2_1_1"><span class="lab"><font color="Green" title="E114">A46</font></span></a>, <a class="txt" href="#E40:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E131">A67</font></span></a>, <a class="txt" href="#E42:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E132">A68</font></span></a>, <a class="txt" href="graph_5.html#E8"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/><a NAME="E45:99_2_1_2_1_1_1_1_2"/>
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c31">t1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">t1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c32">t2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#T30" target="_self" title="GRAPH_5:th.30">Th30</a></span>;<br/><a NAME="E46:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c31">t1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c21">s1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E132">A68</font></span></a>, <a class="ref" href="xboole_1.html#T33" title="XBOOLE_1:th.33">XBOOLE_1:33</a></span>;<br/><a NAME="E47:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A70</font></span>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c31">t1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:99_2_1"><span class="lab"><font color="Green" title="E105">A37</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E48:99_2_1_2_1_1_1_1_2"/>
( <font color="Maroon" title="c31">t1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:99_2_1"><span class="lab"><font color="Green" title="E101">A33</font></span></a>, <a class="txt" href="#E3:99_2_1_2_1_1"><span class="lab"><font color="Green" title="E112">A44</font></span></a>, <a class="txt" href="#E40:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E131">A67</font></span></a>, <a class="txt" href="#E42:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E132">A68</font></span></a>, <a class="txt" href="graph_5.html#E8"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/><a NAME="E49:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c31">t1</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="c9">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E133">A69</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E50:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c31">t1</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="c9">v2</font>,<font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E134">A70</font></span></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/><a NAME="E51:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A71</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c3">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">t1</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>;<br/><a NAME="E52:99_2_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E136">A72</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c32">t2</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="ref" href="graph_5.html#T54" target="_self" title="GRAPH_5:th.54">Th54</a></span>;<br/><a NAME="E53:99_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="c21">s1</font>,<font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c31">t1</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c32">t2</font>,<font color="Maroon" title="c3">W</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="txt" href="#E42:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E132">A68</font></span></a>, <a class="ref" href="graph_5.html#T50" target="_self" title="GRAPH_5:th.50">Th50</a>, <a class="ref" href="graph_5.html#T58" target="_self" title="GRAPH_5:th.58">Th58</a></span>;<br/><a NAME="E54:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c31">t1</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c21">s1</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E136">A72</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E55:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c3">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="c21">s1</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E135">A71</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E56:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A73</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c25">q9</font>,<font color="Maroon" title="c3">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="c21">s1</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E122">A58</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E57:99_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="c27">qx</font>,<font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c25">q9</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c22">s2</font>,<font color="Maroon" title="c3">W</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="ref" href="graph_5.html#T50" target="_self" title="GRAPH_5:th.50">Th50</a>, <a class="ref" href="graph_5.html#T58" target="_self" title="GRAPH_5:th.58">Th58</a></span>;<br/><a NAME="E58:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c27">qx</font>,<font color="Maroon" title="c3">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="c14">s</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:99_2_1"><span class="lab"><font color="Green" title="E110">A42</font></span></a>, <a class="txt" href="#E56:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E137">A73</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E59:99_2_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A74</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c14">s</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E130">A66</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:99_2_1_2_1_1_1_1_2_2"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c7">R</font>,<font color="Maroon" title="c3">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_1_2_1_1_1_1_2_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c33">u</font> be   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c4">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c33">u</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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c33">u</font>,<font color="Maroon" title="c3">W</font> )</span><br/><span class="kw">assume </span><a NAME="E1:99_2_1_2_1_1_1_1_2_2_1"/>
<font color="Maroon" title="c33">u</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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c33">u</font>,<font color="Maroon" title="c3">W</font></span><br/><a NAME="E2:99_2_1_2_1_1_1_1_2_2_1"/><span class="kw">then </span>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c3">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="c33">u</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:99"><span class="lab"><font color="Green" title="E85">A15</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>;<br/><span class="kw">hence </span><a NAME="E3:99_2_1_2_1_1_1_1_2_2_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c33">u</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E138">A74</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:99_2_1_2_1_1_1_1_2_2"/>
<font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:99"><span class="lab"><font color="Green" title="E83">A13</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:99_2_1_2_1_1_1_1_2_3"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font></span><br/><a NAME="E2:99_2_1_2_1_1_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E139">A75</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">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="c14">s</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:99_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E138">A74</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_1_2_1_1_1_1_2_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c33">u</font> be   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c4">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c33">u</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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">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="c33">u</font>,<font color="Maroon" title="c3">W</font> )</span><br/><span class="kw">assume </span><a NAME="E1:99_2_1_2_1_1_1_1_2_3_1"/>
<font color="Maroon" title="c33">u</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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">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="c33">u</font>,<font color="Maroon" title="c3">W</font></span><br/><a NAME="E2:99_2_1_2_1_1_1_1_2_3_1"/><span class="kw">then </span>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c3">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="c33">u</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:99"><span class="lab"><font color="Green" title="E85">A15</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>;<br/><span class="kw">hence </span><a NAME="E3:99_2_1_2_1_1_1_1_2_3_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">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="c33">u</font>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_2_1_2_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E139">A75</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:99_2_1_2_1_1_1_1_2_3"/>
<font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:99"><span class="lab"><font color="Green" title="E86">A16</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:99_2_1_2_1_2"/>
not <font color="Maroon" title="c9">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c21">s1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) &amp; (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) )</span><br/><div class="add"><a NAME="E2:99_2_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A76</font></span>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c21">s1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c21">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T65" title="ZFMISC_1:th.65">ZFMISC_1:65</a></span>;<br/><a NAME="E3:99_2_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c21">s1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:99"><span class="lab"><font color="Green" title="E89">A21</font></span></a>, <a class="txt" href="#E18:99_2_1"><span class="lab"><font color="Green" title="E104">A36</font></span></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/><a NAME="E4:99_2_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c14">s</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:99_2_1"><span class="lab"><font color="Green" title="E104">A36</font></span></a>, <a class="txt" href="#E2:99_2_1_2_1_2"><span class="lab"><font color="Green" title="E112">A76</font></span></a>, <a class="ref" href="xboole_1.html#T43" title="XBOOLE_1:th.43">XBOOLE_1:43</a></span>;<br/><a NAME="E5:99_2_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">s</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="c10">v3</font>,<font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:99"><span class="lab"><font color="Green" title="E90">A22</font></span></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E6:99_2_1_2_1_2"/>
( (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) &amp; (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:99"><span class="lab"><font color="Green" title="E78">A8</font></span></a>, <a class="txt" href="#E20:99"><span class="lab"><font color="Green" title="E87">A17</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_2"><span class="kw">suppose </span></a><a NAME="E1:99_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) &amp; (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) )</span><br/><div class="add"><a NAME="E2:99_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A77</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:99_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A78</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:99"><span class="lab"><font color="Green" title="E92">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:99_2_2"/><span class="lab"><font color="Green" title="E95">A79</font></span>: 
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K1" title="GRAPH_5:func.1">vertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:99"><span class="lab"><font color="Green" title="E92">A24</font></span></a>, <a class="txt" href="#E2:99_2_2"><span class="lab"><font color="Green" title="E93">A77</font></span></a>, <a class="ref" href="graph_5.html#T29" target="_self" title="GRAPH_5:th.29">Th29</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:99_2_2"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c14">s</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c14">s</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:99_2_2_1"/><span class="lab"><font color="Green" title="E96">A80</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c14">s</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font></span><br/>

<a NAME="E2:99_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A81</font></span>: 
not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E3:99_2_2_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K1" title="GRAPH_5:func.1">vertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_2_2"><span class="lab"><font color="Green" title="E95">A79</font></span></a>, <a class="txt" href="#E1:99_2_2_1"><span class="lab"><font color="Green" title="E96">A80</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E4:99_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A82</font></span>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E5:99_2_2_1"/><span class="lab"><font color="Green" title="E99">A83</font></span>: 
<font color="Maroon" title="c14">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:99"><span class="lab"><font color="Green" title="E92">A24</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<font color="Maroon" title="c10">v3</font> = 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">s</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:99"><span class="lab"><font color="Green" title="E90">A22</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>
<br/>.= 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_2_2"><span class="lab"><font color="Green" title="E94">A78</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:99_2_2_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:99"><span class="lab"><font color="Green" title="E90">A22</font></span></a>, <a class="txt" href="#E12:99"><span class="lab"><font color="Green" title="E82">A12</font></span></a>, <a class="txt" href="#E2:99_2_2_1"><span class="lab"><font color="Green" title="E97">A81</font></span></a>, <a class="txt" href="#E4:99_2_2_1"><span class="lab"><font color="Green" title="E98">A82</font></span></a>, <a class="txt" href="#E5:99_2_2_1"><span class="lab"><font color="Green" title="E99">A83</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:99_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">s</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="c10">v3</font>,<font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:99"><span class="lab"><font color="Green" title="E90">A22</font></span></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:99_2_2"/>
( (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">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="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c6">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) &amp; (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c7">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="c10">v3</font>,<font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">W</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:99"><span class="lab"><font color="Green" title="E78">A8</font></span></a>, <a class="txt" href="#E20:99"><span class="lab"><font color="Green" title="E87">A17</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
