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<b>let </b><font color="Maroon" title="c1">W</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">G</font> being <a href="graph_1.html#V7" title="GRAPH_1:attr.7">finite</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a><br/> for <font color="Olive" title="b3">P</font>, <font color="Olive" title="b4">Q</font> being <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Olive" title="b2">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="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Olive" title="b2">G</font>  st <font color="Maroon" title="c1">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; <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="b5">v1</font>,<font color="Olive" title="b6">v2</font>,<font color="Olive" title="b1">V</font>,<font color="Maroon" title="c1">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">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="Olive" title="b1">V</font>,<font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b8">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Olive" title="b2">G</font> or  not <font color="Olive" title="b8">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="b3">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a> <font color="Olive" title="b1">V</font>,<font color="Olive" title="b5">v1</font>,<font color="Maroon" title="c1">W</font> holds <br/><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="b5">v1</font>,<font color="Olive" title="b7">v3</font>,<font color="Olive" title="b1">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="c1">W</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font> being <a href="graph_1.html#V7" title="GRAPH_1:attr.7">finite</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a><br/> for <font color="Olive" title="b2">P</font>, <font color="Olive" title="b3">Q</font> being <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Olive" title="b1">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="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Olive" title="b1">G</font>  st <font color="Maroon" title="c1">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; <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="b4">v1</font>,<font color="Olive" title="b5">v2</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">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">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="c1">W</font> &amp; ( for <font color="Olive" title="b7">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Olive" title="b1">G</font> or  not <font color="Olive" title="b7">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="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="b4">v1</font>,<font color="Maroon" title="c1">W</font> holds <br/><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="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="c1">W</font></span><br/><b>let </b><font color="Maroon" title="c3">G</font> be  <a href="graph_1.html#V7" title="GRAPH_1:attr.7">finite</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font> being <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c3">G</font><br/> for <font color="Olive" title="b3">v1</font>, <font color="Olive" title="b4">v2</font>, <font color="Olive" title="b5">v3</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c3">G</font>  st <font color="Maroon" title="c1">W</font> <a href="graph_5.html#R5" title="GRAPH_5:pred.5">is_weight&gt;=0of</a> <font color="Maroon" title="c3">G</font> &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="b3">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">W</font> &amp; <font color="Olive" title="b3">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">v2</font> &amp; <font color="Olive" title="b3">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">v3</font> &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="b3">v1</font>,<font color="Olive" title="b5">v3</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b6">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G</font> or  not <font color="Olive" title="b6">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Olive" title="b4">v2</font>,<font color="Olive" title="b5">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="b3">v1</font>,<font color="Maroon" title="c1">W</font> holds <br/><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="b3">v1</font>,<font color="Olive" title="b5">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="b4">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c1">W</font></span><br/><b>let </b><font color="Maroon" title="c4">P</font>, <font color="Maroon" title="c5">Q</font> be  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c3">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c3">G</font>  st <font color="Maroon" title="c1">W</font> <a href="graph_5.html#R5" title="GRAPH_5:pred.5">is_weight&gt;=0of</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c4">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="c1">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="c5">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="c1">W</font> &amp; ( for <font color="Olive" title="b4">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G</font> or  not <font color="Olive" title="b4">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="c4">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="c1">W</font> holds <br/><font color="Maroon" title="c5">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="c1">W</font></span><br/><b>let </b><font color="Maroon" title="c6">v1</font>, <font color="Maroon" title="c7">v2</font>, <font color="Maroon" title="c8">v3</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c3">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">W</font> <a href="graph_5.html#R5" title="GRAPH_5:pred.5">is_weight&gt;=0of</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c4">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c6">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v2</font> &amp; <font color="Maroon" title="c6">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v3</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">v3</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G</font> or  not <font color="Olive" title="b1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c8">v3</font> ) ) &amp; <font color="Maroon" title="c4">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="c6">v1</font>,<font color="Maroon" title="c1">W</font> implies <font color="Maroon" title="c5">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">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="c7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c1">W</font> )</span><br/>















<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c1">W</font> <a href="graph_5.html#R5" title="GRAPH_5:pred.5">is_weight&gt;=0of</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c4">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c6">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v2</font> &amp; <font color="Maroon" title="c6">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v3</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">v3</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G</font> or  not <font color="Olive" title="b1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c8">v3</font> ) ) &amp; <font color="Maroon" title="c4">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="c6">v1</font>,<font color="Maroon" title="c1">W</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">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="c7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c1">W</font></span><br/>

<b>set </b><font color="Maroon" title="c9">V'</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="c7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c10">FS</font> = the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G</font>;<br/>
<b>set </b><font color="Maroon" title="c11">FT</font> = the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G</font>;<br/>
<a NAME="E2:17"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c1">W</font> <a href="graph_5.html#R6" title="GRAPH_5:pred.6">is_weight_of</a> <font color="Maroon" title="c3">G</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="graph_5.html#T50" title="GRAPH_5:th.50">GRAPH_5:50</a></i>;<br/>
<div><i><font color="Green" title="E20">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">S</font> be  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c3">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">S</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">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="c7">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="c5">Q</font>,<font color="Maroon" title="c1">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">b<sub>1</sub></font>,<font color="Maroon" title="c1">W</font> )</span><br/><b>assume </b><a NAME="E1:17_1"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c12">S</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">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="c7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c1">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">b<sub>1</sub></font>,<font color="Maroon" title="c1">W</font></span><br/><b>then consider </b><font color="Maroon" title="c13">s</font> being  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c3">G</font><b> such that </b><br/><a NAME="E3:17_1"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c13">s</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">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="c7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="graph_5.html#T66" title="GRAPH_5:th.66">GRAPH_5:66</a></i>;<br/><a NAME="E4:17_1"/><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c13">s</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">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="c7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:17_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>;<br/><a NAME="E5:17_1"/><b>then </b><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon" title="c13">s</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">v3</font>
 <b>by </b><i><a class="ref" href="graph_5.html#D4" title="GRAPH_5:def.4">GRAPH_5:def 4</a></i>;<br/><a NAME="E6:17_1"/><b>then </b>
<font color="Maroon" title="c13">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> 
 <b>by </b><i><a class="ref" href="graph_5.html#D3" title="GRAPH_5:def.3">GRAPH_5:def 3</a></i>;<br/><a NAME="E7:17_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">s</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:17_1"/><i><font color="Green" title="E25">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">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> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">s1</font>, <font color="Maroon" title="c16">s2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E11:17_1"/><i><font color="Green" title="E26">A9</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c16">s2</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E9:17_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="finseq_2.html#T25" title="FINSEQ_2:th.25">FINSEQ_2:25</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">s1</font> = <font color="Maroon" title="c15">s1</font>, <font color="Maroon" title="c18">s2</font> = <font color="Maroon" title="c16">s2</font> as  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c3">G</font> <b>by </b><i><a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="graph_5.html#T17" title="GRAPH_5:th.17">GRAPH_5:17</a></i>;<br/><a NAME="E13:17_1"/><i><font color="Green" title="E27">A10</font></i>: 
<font color="Maroon" title="c18">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="c13">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="c13">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E9:17_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="graphsp.html#E7"><i><font color="Green" title="E7">Lm2</font></i></a></i>;<br/><a NAME="E14:17_1"/><i><font color="Green" title="E28">A11</font></i>: 
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c6">v1</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c13">s</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v3</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E5:17_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="graph_5.html#D3" title="GRAPH_5:def.3">GRAPH_5:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">vx</font> = the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c3">G</font> <b>by </b><i><a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="graphsp.html#E13"><i><font color="Green" title="E13">Lm3</font></i></a></i>;<br/><div><i><font color="Green" title="E29">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_1"/>
<font color="Maroon" title="c19">vx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:17_1_1"/><b>then </b><i><font color="Green" title="E30">A13</font></i>: 
<font color="Maroon" title="c18">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c8">v3</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E13:17_1"><i><font color="Green" title="E27">A10</font></i></a>, <a class="txt" href="graphsp.html#E14:17_1"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></i>;<br/><a NAME="E3:17_1_1"/>
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="c18">s2</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="graphsp.html#E2:17_1_1"><i><font color="Green" title="E30">A13</font></i></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E17:17_1"/><i><font color="Green" title="E30">A14</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c13">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="c8">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="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="c7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="graphsp.html#E4:17_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="graph_5.html#D4" title="GRAPH_5:def.4">GRAPH_5:def 4</a></i>;<br/><a NAME="E18:17_1"/><i><font color="Green" title="E31">A15</font></i>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c1">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="c12">S</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:17_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="graphsp.html#E3:17_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_2"/>
(  not <font color="Maroon" title="c7">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="c13">s</font> or <font color="Maroon" title="c7">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v3</font> or ( <font color="Maroon" title="c7">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="c13">s</font> &amp; <font color="Maroon" title="c7">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v3</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_1"><b>suppose </b></a><a NAME="E1:17_1_2_1"/><i><font color="Green" title="E32">A16</font></i>: 
(  not <font color="Maroon" title="c7">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="c13">s</font> or <font color="Maroon" title="c7">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c1">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">b<sub>1</sub></font>,<font color="Maroon" title="c1">W</font></span><br/><div class="add"><b>set </b><font color="Maroon" title="c20">Vs</font> =  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c13">s</font>;<br/><a NAME="E2:17_1_2_1"/>
<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="c13">s</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c7">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>
 <b>by </b><i><a class="txt" href="graphsp.html#E17:17_1"><i><font color="Green" title="E30">A14</font></i></a>, <a class="ref" href="xboole_1.html#T43" title="XBOOLE_1:th.43">XBOOLE_1:43</a></i>;<br/><a NAME="E3:17_1_2_1"/><b>then </b><i><font color="Green" title="E33">A17</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_2_1_1_1"/>
(  not <font color="Maroon" title="c7">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="c13">s</font> or <font color="Maroon" title="c7">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v3</font> )
 </span><b>by </b><i><a class="txt" href="graphsp.html#E1:17_1_2_1"><i><font color="Green" title="E32">A16</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_2_1_1_1_1"/><i><font color="Green" title="E34">A18</font></i>: 
not <font color="Maroon" title="c7">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="c13">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c13">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="c8">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><br/><div class="add"><a NAME="E2:17_1_2_1_1_1_1"/>
<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="c13">s</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c8">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>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:17_1_2_1"><i><font color="Green" title="E33">A17</font></i></a>, <a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a></i>;<br/><b>hence </b><a NAME="E3:17_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c13">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="c8">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>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:17_1_2_1_1_1_1"><i><font color="Green" title="E34">A18</font></i></a>, <a class="ref" href="zfmisc_1.html#T65" title="ZFMISC_1:th.65">ZFMISC_1:65</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_2_1_1_1_2"/>
<font color="Maroon" title="c7">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c13">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="c8">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><br/><div class="add"><b>hence </b><a NAME="E2:17_1_2_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="c13">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="c8">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>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:17_1_2_1"><i><font color="Green" title="E33">A17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E5:17_1_2_1"/><b>then </b>
<font color="Maroon" title="c13">s</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">v3</font>,<font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:17_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="graph_5.html#D4" title="GRAPH_5:def.4">GRAPH_5:def 4</a></i>;<br/><a NAME="E6:17_1_2_1"/><b>then </b>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c1">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="c13">s</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>;<br/><b>hence </b><a NAME="E7:17_1_2_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c1">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="c12">S</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E18:17_1"><i><font color="Green" title="E31">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_2"><b>suppose </b></a><a NAME="E1:17_1_2_2"/><i><font color="Green" title="E32">A19</font></i>: 
( <font color="Maroon" title="c7">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="c13">s</font> &amp; <font color="Maroon" title="c7">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c1">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">b<sub>1</sub></font>,<font color="Maroon" title="c1">W</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c20">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:17_1_2_2"/><i><font color="Green" title="E33">A20</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j</font> &amp; <font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">s</font> &amp; <font color="Maroon" title="c7">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="graphsp.html#E14:17_1"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="graph_5.html#T32" title="GRAPH_5:th.32">GRAPH_5:32</a></i>;<br/><a NAME="E4:17_1_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="graphsp.html#E9:17_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="graphsp.html#E14:17_1"><i><font color="Green" title="E28">A11</font></i></a>, <a class="txt" href="graphsp.html#E1:17_1_2_2"><i><font color="Green" title="E32">A19</font></i></a>, <a class="txt" href="graphsp.html#E3:17_1_2_2"><i><font color="Green" title="E33">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:17_1_2_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E6:17_1_2_2"/><b>then </b><i><font color="Green" title="E34">A21</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">s1</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>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E7:17_1_2_2"/><i><font color="Green" title="E35">A22</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">s</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E9:17_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><i><font color="Green" title="E36">A23</font></i>: <font color="Maroon" title="c19">vx</font> = 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c17">s1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="graphsp.html#E7"><i><font color="Green" title="E7">Lm2</font></i></a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c17">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E6:17_1_2_2"><i><font color="Green" title="E34">A21</font></i></a>, <a class="txt" href="graphsp.html#E7:17_1_2_2"><i><font color="Green" title="E35">A22</font></i></a>, <a class="ref" href="graph_1.html#D12" title="GRAPH_1:def.12">GRAPH_1:def 12</a></i>
;<br/><div><i><font color="Green" title="E37">A24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_2_2_2"/>
<font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">s1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:17_1_2_2_2"/><b>then </b>
<font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">s1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:17_1_2_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E9:17_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="graphsp.html#E14:17_1"><i><font color="Green" title="E28">A11</font></i></a>, <a class="txt" href="graphsp.html#E1:17_1_2_2"><i><font color="Green" title="E32">A19</font></i></a>, <a class="txt" href="graphsp.html#E3:17_1_2_2"><i><font color="Green" title="E33">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c21">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E11:17_1_2_2"/><i><font color="Green" title="E38">A25</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>consider </b><font color="Maroon" title="c22">t1</font>, <font color="Maroon" title="c23">t2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E13:17_1_2_2"/><i><font color="Green" title="E39">A26</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">k</font> &amp; <font color="Maroon" title="c17">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">t1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c23">t2</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E11:17_1_2_2"><i><font color="Green" title="E38">A25</font></i></a>, <a class="ref" href="finseq_2.html#T25" title="FINSEQ_2:th.25">FINSEQ_2:25</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">t1</font> = <font color="Maroon" title="c22">t1</font>, <font color="Maroon" title="c25">t2</font> = <font color="Maroon" title="c23">t2</font> as  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c3">G</font> <b>by </b><i><a class="txt" href="graphsp.html#E13:17_1_2_2"><i><font color="Green" title="E39">A26</font></i></a>, <a class="ref" href="graph_5.html#T17" title="GRAPH_5:th.17">GRAPH_5:17</a></i>;<br/><i><font color="Green" title="E40">A27</font></i>: the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> = 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E3:17_1_2_2"><i><font color="Green" title="E33">A20</font></i></a>, <a class="txt" href="graphsp.html#E13:17_1_2_2"><i><font color="Green" title="E39">A26</font></i></a>, <a class="txt" href="graphsp.html#E6"><i><font color="Green" title="E6">Lm1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">v1</font>
<b>by </b><i><a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="graphsp.html#E14:17_1"><i><font color="Green" title="E28">A11</font></i></a>, <a class="txt" href="graphsp.html#E6:17_1_2_2"><i><font color="Green" title="E34">A21</font></i></a>, <a class="txt" href="graphsp.html#E6"><i><font color="Green" title="E6">Lm1</font></i></a></i>
;<br/><a NAME="E16:17_1_2_2"/><i><font color="Green" title="E41">A28</font></i>: 
<font color="Maroon" title="c24">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> 
 <b>by </b><i><a class="txt" href="graphsp.html#E3:17_1_2_2"><i><font color="Green" title="E33">A20</font></i></a>, <a class="txt" href="graphsp.html#E13:17_1_2_2"><i><font color="Green" title="E39">A26</font></i></a>, <a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></i>;<br/>the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">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="c24">t1</font></span>)</span></span>)</span> = 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E3:17_1_2_2"><i><font color="Green" title="E33">A20</font></i></a>, <a class="txt" href="graphsp.html#E13:17_1_2_2"><i><font color="Green" title="E39">A26</font></i></a>, <a class="txt" href="graphsp.html#E6"><i><font color="Green" title="E6">Lm1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">v2</font>
<b>by </b><i><a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="graphsp.html#E3:17_1_2_2"><i><font color="Green" title="E33">A20</font></i></a>, <a class="txt" href="graphsp.html#E9:17_1_2_2"><i><font color="Green" title="E37">A24</font></i></a>, <a class="txt" href="graphsp.html#E6"><i><font color="Green" title="E6">Lm1</font></i></a></i>
;<br/><a NAME="E18:17_1_2_2"/><b>then </b><i><font color="Green" title="E42">A29</font></i>: 
<font color="Maroon" title="c24">t1</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E15:17_1_2_2"><i><font color="Green" title="E40">A27</font></i></a>, <a class="txt" href="graphsp.html#E16:17_1_2_2"><i><font color="Green" title="E41">A28</font></i></a>, <a class="ref" href="graph_5.html#D3" title="GRAPH_5:def.3">GRAPH_5:def 3</a></i>;<br/><a NAME="E19:17_1_2_2"/><i><font color="Green" title="E43">A30</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">s2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a></i>;<br/><a NAME="E20:17_1_2_2"/><b>then </b><i><font color="Green" title="E44">A31</font></i>: 
not <font color="Maroon" title="c8">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="c17">s1</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="graphsp.html#E14:17_1"><i><font color="Green" title="E28">A11</font></i></a>, <a class="txt" href="graphsp.html#E6:17_1_2_2"><i><font color="Green" title="E34">A21</font></i></a>, <a class="ref" href="graph_5.html#T21" title="GRAPH_5:th.21">GRAPH_5:21</a></i>;<br/><b>set </b><font color="Maroon" title="c26">Vt</font> =  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c24">t1</font>;<br/><a NAME="E21:17_1_2_2"/>
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c24">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="c24">t1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">t2</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_5.html#T30" title="GRAPH_5:th.30">GRAPH_5:30</a></i>;<br/><a NAME="E22:17_1_2_2"/><b>then </b><i><font color="Green" title="E45">A32</font></i>: 
not <font color="Maroon" title="c8">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="c24">t1</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="graphsp.html#E14:17_1"><i><font color="Green" title="E28">A11</font></i></a>, <a class="txt" href="graphsp.html#E6:17_1_2_2"><i><font color="Green" title="E34">A21</font></i></a>, <a class="txt" href="graphsp.html#E13:17_1_2_2"><i><font color="Green" title="E39">A26</font></i></a>, <a class="txt" href="graphsp.html#E19:17_1_2_2"><i><font color="Green" title="E43">A30</font></i></a>, <a class="ref" href="graph_5.html#T21" title="GRAPH_5:th.21">GRAPH_5:21</a></i>;<br/><a NAME="E23:17_1_2_2"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c18">s2</font></span>)</span></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="c8">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="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="c7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="graphsp.html#E4:17_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="graph_5.html#D4" title="GRAPH_5:def.4">GRAPH_5:def 4</a></i>;<br/><a NAME="E24:17_1_2_2"/><b>then </b><i><font color="Green" title="E46">A33</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">t1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">t2</font></span>)</span></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="c8">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="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="c7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="graphsp.html#E13:17_1_2_2"><i><font color="Green" title="E39">A26</font></i></a>, <a class="ref" href="graph_5.html#T26" title="GRAPH_5:th.26">GRAPH_5:26</a></i>;<br/><a NAME="E25:17_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c24">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="c8">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="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="c7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="graph_5.html#T26" title="GRAPH_5:th.26">GRAPH_5:26</a></i>;<br/><a NAME="E26:17_1_2_2"/><b>then </b>
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c24">t1</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="c7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="graphsp.html#E22:17_1_2_2"><i><font color="Green" title="E45">A32</font></i></a>, <a class="ref" href="zfmisc_1.html#T65" title="ZFMISC_1:th.65">ZFMISC_1:65</a></i>;<br/><a NAME="E27:17_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c24">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="c7">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>
 <b>by </b><i><a class="ref" href="xboole_1.html#T43" title="XBOOLE_1:th.43">XBOOLE_1:43</a></i>;<br/><a NAME="E28:17_1_2_2"/><b>then </b>
<font color="Maroon" title="c24">t1</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E18:17_1_2_2"><i><font color="Green" title="E42">A29</font></i></a>, <a class="ref" href="graph_5.html#D4" title="GRAPH_5:def.4">GRAPH_5:def 4</a></i>;<br/><a NAME="E29:17_1_2_2"/><b>then </b><i><font color="Green" title="E47">A34</font></i>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c1">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">t1</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>;<br/><a NAME="E30:17_1_2_2"/>
<font color="Maroon" title="c19">vx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">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="c17">s1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="graphsp.html#E6:17_1_2_2"><i><font color="Green" title="E34">A21</font></i></a>, <a class="txt" href="graphsp.html#E8:17_1_2_2"><i><font color="Green" title="E36">A23</font></i></a>, <a class="txt" href="graphsp.html#E6"><i><font color="Green" title="E6">Lm1</font></i></a></i>;<br/><a NAME="E31:17_1_2_2"/><b>then </b><i><font color="Green" title="E48">A35</font></i>: 
<font color="Maroon" title="c19">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="c17">s1</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E6:17_1_2_2"><i><font color="Green" title="E34">A21</font></i></a>, <a class="txt" href="graphsp.html#E15"><i><font color="Green" title="E15">Lm4</font></i></a></i>;<br/><a NAME="E32:17_1_2_2"/>
not <font color="Maroon" title="c19">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="c7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="graphsp.html#E16:17_1"><i><font color="Green" title="E29">A12</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E33:17_1_2_2"/><b>then </b><i><font color="Green" title="E49">A36</font></i>: 
<font color="Maroon" title="c19">vx</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="c17">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="c7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="graphsp.html#E31:17_1_2_2"><i><font color="Green" title="E48">A35</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E34:17_1_2_2"/>
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c17">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="c7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="graphsp.html#E13:17_1_2_2"><i><font color="Green" title="E39">A26</font></i></a>, <a class="txt" href="graphsp.html#E20:17_1_2_2"><i><font color="Green" title="E44">A31</font></i></a>, <a class="txt" href="graphsp.html#E24:17_1_2_2"><i><font color="Green" title="E46">A33</font></i></a>, <a class="ref" href="zfmisc_1.html#T65" title="ZFMISC_1:th.65">ZFMISC_1:65</a></i>;<br/><a NAME="E35:17_1_2_2"/><b>then </b><i><font color="Green" title="E50">A37</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c17">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="c7">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>
 <b>by </b><i><a class="ref" href="xboole_1.html#T43" title="XBOOLE_1:th.43">XBOOLE_1:43</a></i>;<br/><a NAME="E36:17_1_2_2"/>
not <font color="Maroon" title="c6">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="c18">s2</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="graphsp.html#E14:17_1"><i><font color="Green" title="E28">A11</font></i></a>, <a class="txt" href="graphsp.html#E6:17_1_2_2"><i><font color="Green" title="E34">A21</font></i></a>, <a class="ref" href="graph_5.html#T21" title="GRAPH_5:th.21">GRAPH_5:21</a></i>;<br/><a NAME="E37:17_1_2_2"/><b>then </b>
<font color="Maroon" title="c19">vx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v1</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="graphsp.html#E15"><i><font color="Green" title="E15">Lm4</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c27">q</font> being  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c3">G</font><b> such that </b><br/><a NAME="E39:17_1_2_2"/><i><font color="Green" title="E51">A38</font></i>: 
( <font color="Maroon" title="c27">q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c19">vx</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c27">q</font>,<font color="Maroon" title="c1">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="c4">P</font>,<font color="Maroon" title="c1">W</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="graphsp.html#E33:17_1_2_2"><i><font color="Green" title="E49">A36</font></i></a>, <a class="txt" href="graphsp.html#E35:17_1_2_2"><i><font color="Green" title="E50">A37</font></i></a>, <a class="ref" href="graph_5.html#D19" title="GRAPH_5:def.19">GRAPH_5:def 19</a></i>;<br/><a NAME="E40:17_1_2_2"/><i><font color="Green" title="E52">A39</font></i>: 
<font color="Maroon" title="c27">q</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c19">vx</font>,<font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E39:17_1_2_2"><i><font color="Green" title="E51">A38</font></i></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>;<br/><a NAME="E41:17_1_2_2"/><i><font color="Green" title="E53">A40</font></i>: 
<font color="Maroon" title="c18">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c19">vx</font>,<font color="Maroon" title="c8">v3</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E13:17_1"><i><font color="Green" title="E27">A10</font></i></a>, <a class="txt" href="graphsp.html#E14:17_1"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></i>;<br/><a NAME="E42:17_1_2_2"/><i><font color="Green" title="E54">A41</font></i>: 
<font color="Maroon" title="c27">q</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c19">vx</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E40:17_1_2_2"><i><font color="Green" title="E52">A39</font></i></a>, <a class="ref" href="graph_5.html#D4" title="GRAPH_5:def.4">GRAPH_5:def 4</a></i>;<br/><a NAME="E43:17_1_2_2"/><b>then </b>
<font color="Maroon" title="c27">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="graph_5.html#D3" title="GRAPH_5:def.3">GRAPH_5:def 3</a></i>;<br/><a NAME="E44:17_1_2_2"/><b>then </b><i><font color="Green" title="E55">A42</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">q</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></i>;<br/><b>then consider </b><font color="Maroon" title="c28">p</font> being  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c3">G</font><b> such that </b><br/><a NAME="E46:17_1_2_2"/><i><font color="Green" title="E56">A43</font></i>: 
( <font color="Maroon" title="c28">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c18">s2</font> &amp; <font color="Maroon" title="c28">p</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">v3</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="graphsp.html#E41:17_1_2_2"><i><font color="Green" title="E53">A40</font></i></a>, <a class="txt" href="graphsp.html#E42:17_1_2_2"><i><font color="Green" title="E54">A41</font></i></a>, <a class="ref" href="graph_5.html#T37" title="GRAPH_5:th.37">GRAPH_5:37</a></i>;<br/><a NAME="E47:17_1_2_2"/>
<font color="Maroon" title="c28">p</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">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="c19">vx</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="graphsp.html#E40:17_1_2_2"><i><font color="Green" title="E52">A39</font></i></a>, <a class="txt" href="graphsp.html#E41:17_1_2_2"><i><font color="Green" title="E53">A40</font></i></a>, <a class="txt" href="graphsp.html#E44:17_1_2_2"><i><font color="Green" title="E55">A42</font></i></a>, <a class="txt" href="graphsp.html#E46:17_1_2_2"><i><font color="Green" title="E56">A43</font></i></a>, <a class="ref" href="graph_5.html#T38" title="GRAPH_5:th.38">GRAPH_5:38</a></i>;<br/><a NAME="E48:17_1_2_2"/><b>then </b>
<font color="Maroon" title="c28">p</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">v3</font>,<font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E33:17_1_2_2"><i><font color="Green" title="E49">A36</font></i></a>, <a class="txt" href="graphsp.html#E35:17_1_2_2"><i><font color="Green" title="E50">A37</font></i></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>;<br/><a NAME="E49:17_1_2_2"/><b>then </b>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c1">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="c28">p</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>;<br/><a NAME="E50:17_1_2_2"/><b>then </b><i><font color="Green" title="E57">A44</font></i>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c1">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c27">q</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c18">s2</font>,<font color="Maroon" title="c1">W</font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="graphsp.html#E46:17_1_2_2"><i><font color="Green" title="E56">A43</font></i></a>, <a class="ref" href="graph_5.html#T58" title="GRAPH_5:th.58">GRAPH_5:58</a></i>;<br/><a NAME="E51:17_1_2_2"/><i><font color="Green" title="E58">A45</font></i>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c27">q</font>,<font color="Maroon" title="c1">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">t1</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E29:17_1_2_2"><i><font color="Green" title="E47">A34</font></i></a>, <a class="txt" href="graphsp.html#E39:17_1_2_2"><i><font color="Green" title="E51">A38</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E52:17_1_2_2"/><i><font color="Green" title="E59">A46</font></i>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c17">s1</font>,<font color="Maroon" title="c1">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="c24">t1</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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">t2</font>,<font color="Maroon" title="c1">W</font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="graphsp.html#E13:17_1_2_2"><i><font color="Green" title="E39">A26</font></i></a>, <a class="ref" href="graph_5.html#T58" title="GRAPH_5:th.58">GRAPH_5:58</a></i>;<br/><a NAME="E53:17_1_2_2"/>
 <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="c25">t2</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="graph_5.html#T54" title="GRAPH_5:th.54">GRAPH_5:54</a></i>;<br/><a NAME="E54:17_1_2_2"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c24">t1</font>,<font color="Maroon" title="c1">W</font></span>)</span> <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="c17">s1</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E52:17_1_2_2"><i><font color="Green" title="E59">A46</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E55:17_1_2_2"/><b>then </b><i><font color="Green" title="E60">A47</font></i>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c27">q</font>,<font color="Maroon" title="c1">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="c17">s1</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E51:17_1_2_2"><i><font color="Green" title="E58">A45</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E56:17_1_2_2"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c1">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="c17">s1</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c18">s2</font>,<font color="Maroon" title="c1">W</font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="graphsp.html#E11:17_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="graph_5.html#T58" title="GRAPH_5:th.58">GRAPH_5:58</a></i>;<br/><a NAME="E57:17_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c27">q</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c18">s2</font>,<font color="Maroon" title="c1">W</font></span>)</span> <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="c13">s</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E55:17_1_2_2"><i><font color="Green" title="E60">A47</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E58:17_1_2_2"/><b>then </b>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c1">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="c13">s</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E50:17_1_2_2"><i><font color="Green" title="E57">A44</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E59:17_1_2_2"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c1">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="c12">S</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E18:17_1"><i><font color="Green" title="E31">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:17"/>
<font color="Maroon" title="c5">Q</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">v3</font>,<font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>;<br/>
<a NAME="E5:17"/><b>then </b>
<font color="Maroon" title="c5">Q</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">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="c7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="graph_5.html#T36" title="GRAPH_5:th.36">GRAPH_5:36</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E6:17"/>
<font color="Maroon" title="c5">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">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="c7">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:17"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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