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<span class="kw">let </span><font color="Maroon" title="c1">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#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> 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="b4">v1</font>, <font color="Olive" title="b5">v2</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="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; <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="b4">v1</font>,<font color="Olive" title="b5">v2</font>,<font color="Maroon" title="c1">V</font>,<font color="Olive" title="b1">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; ( for <font color="Olive" title="b6">Q</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="b7">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 not <font color="Olive" title="b7">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Olive" title="b6">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="b7">v3</font>,<font color="Maroon" title="c1">V</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="b3">P</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="b6">Q</font>,<font color="Olive" title="b1">W</font> ) holds <br/><font color="Olive" title="b3">P</font> <a href="graph_5.html#R7" title="GRAPH_5:pred.7">is_shortestpath_of</a> <font color="Olive" title="b4">v1</font>,<font color="Olive" title="b5">v2</font>,<font color="Olive" title="b1">W</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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#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> 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="b3">v1</font>, <font color="Olive" title="b4">v2</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="c2">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="b3">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">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; ( for <font color="Olive" title="b5">Q</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="b6">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 not <font color="Olive" title="b6">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Olive" title="b5">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="b6">v3</font>,<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b2">P</font>,<font color="Maroon" title="c2">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">Q</font>,<font color="Maroon" title="c2">W</font> ) holds <br/><font color="Olive" title="b2">P</font> <a href="graph_5.html#R7" title="GRAPH_5:pred.7">is_shortestpath_of</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c2">W</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">G</font> be   <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> 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="c3">G</font><br/> for <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font>  st <font color="Maroon" title="c2">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="b2">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</font> &amp; <font color="Olive" title="b2">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">v2</font> &amp; ( for <font color="Olive" title="b4">Q</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="c3">G</font><br/> for <font color="Olive" title="b5">v3</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font>  st not <font color="Olive" title="b5">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">V</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="b2">v1</font>,<font color="Olive" title="b5">v3</font>,<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b1">P</font>,<font color="Maroon" title="c2">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">Q</font>,<font color="Maroon" title="c2">W</font> ) holds <br/><font color="Olive" title="b1">P</font> <a href="graph_5.html#R7" title="GRAPH_5:pred.7">is_shortestpath_of</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c2">W</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">P</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="c3">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> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font>  st <font color="Maroon" title="c2">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="c1">V</font>,<font color="Maroon" title="c2">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; ( for <font color="Olive" title="b3">Q</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="c3">G</font><br/> for <font color="Olive" title="b4">v3</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font>  st not <font color="Olive" title="b4">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">V</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="b1">v1</font>,<font color="Olive" title="b4">v3</font>,<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">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">Q</font>,<font color="Maroon" title="c2">W</font> ) holds <br/><font color="Maroon" title="c4">P</font> <a href="graph_5.html#R7" title="GRAPH_5:pred.7">is_shortestpath_of</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font>,<font color="Maroon" title="c2">W</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">v1</font>, <font color="Maroon" title="c6">v2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">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="c2">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="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</font> &amp; <font color="Maroon" title="c5">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v2</font> &amp; ( for <font color="Olive" title="b1">Q</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="c3">G</font><br/> for <font color="Olive" title="b2">v3</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font>  st not <font color="Olive" title="b2">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Olive" title="b1">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Olive" title="b2">v3</font>,<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">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="b1">Q</font>,<font color="Maroon" title="c2">W</font> ) implies <font color="Maroon" title="c4">P</font> <a href="graph_5.html#R7" title="GRAPH_5:pred.7">is_shortestpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c2">W</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:96"/><span class="lab"><font color="Green" title="E70">A1</font></span>: 
<font color="Maroon" title="c2">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>
 <span class="kw">and </span><br/><a NAME="E2:96"/><span class="lab"><font color="Green" title="E71">A2</font></span>: 
<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="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</font>
 <span class="kw">and </span><br/><a NAME="E3:96"/><span class="lab"><font color="Green" title="E72">A3</font></span>: 
<font color="Maroon" title="c5">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v2</font>
 <span class="kw">and </span><br/><a NAME="E4:96"/><span class="lab"><font color="Green" title="E73">A4</font></span>: 
for <font color="Olive" title="b1">Q</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="c3">G</font><br/> for <font color="Olive" title="b2">v3</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font>  st not <font color="Olive" title="b2">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Olive" title="b1">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Olive" title="b2">v3</font>,<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">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="b1">Q</font>,<font color="Maroon" title="c2">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="c4">P</font> <a href="graph_5.html#R7" title="GRAPH_5:pred.7">is_shortestpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c2">W</font></span><br/>

<a NAME="E5:96"/><span class="lab"><font color="Green" title="E74">A5</font></span>: 
<font color="Maroon" title="c4">P</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>;<br/>
<a NAME="E6:96"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A6</font></span>: 
<font color="Maroon" title="c5">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="ref" href="graph_5.html#T35" target="_self" title="GRAPH_5:th.35">Th35</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E76">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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="c3">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="c7">r</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font> implies  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">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="c2">W</font> )</span><br/><span class="kw">assume </span><a NAME="E1:96_1"/><span class="lab"><font color="Green" title="E77">A8</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">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="c4">P</font>,<font color="Maroon" title="c2">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="c2">W</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_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="E1:96_1_1"/>
(  not  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c7">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font> or  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c7">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_1"/><span class="lab"><font color="Green" title="E78">A9</font></span>: 
not  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c7">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</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="c4">P</font>,<font color="Maroon" title="c2">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="c2">W</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c8">FT</font> = the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c3">G</font>;<br/><span class="kw">set </span><font color="Maroon" title="c9">FS</font> = the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c3">G</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">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> , <font color="Maroon" title="c11">s</font>, <font color="Maroon" title="c12">t</font> being    <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="c3">G</font><span class="kw"> such that </span><br/><a NAME="E3:96_1_1_1"/><span class="lab"><font color="Green" title="E79">A10</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">r</font>
 <span class="kw">and </span><br/><a NAME="E4:96_1_1_1"/><span class="lab"><font color="Green" title="E80">A11</font></span>: 
not  <a href="graph_5.html#K1" title="GRAPH_5:func.1">vertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font>
 <span class="kw">and </span><br/><a NAME="E5:96_1_1_1"/><span class="lab"><font color="Green" title="E81">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i</font>
 <span class="kw">and </span><br/><a NAME="E6:96_1_1_1"/><span class="lab"><font color="Green" title="E82">A13</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c12">t</font>
 <span class="kw">and </span><br/><a NAME="E7:96_1_1_1"/><span class="lab"><font color="Green" title="E83">A14</font></span>: 
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c11">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_1_1"><span class="lab"><font color="Green" title="E78">A9</font></span></a>, <a class="ref" href="graph_5.html#T23" target="_self" title="GRAPH_5:th.23">Th23</a></span>;<br/><a NAME="E8:96_1_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_1_1_1"><span class="lab"><font color="Green" title="E79">A10</font></span></a>, <a class="txt" href="#E6:96_1_1_1"><span class="lab"><font color="Green" title="E82">A13</font></span></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/><a NAME="E9:96_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A15</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_1_1_1"><span class="lab"><font color="Green" title="E81">A12</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">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="E11:96_1_1_1"/><span class="lab"><font color="Green" title="E85">A16</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">t</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="c13">j</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/><span class="kw">reconsider </span><font color="Maroon" title="c14">s</font> = <font color="Maroon" title="c11">s</font>, <font color="Maroon" title="c15">t</font> = <font color="Maroon" title="c12">t</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="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:96_1_1_1"><span class="lab"><font color="Green" title="E82">A13</font></span></a>, <a class="ref" href="graph_5.html#T14" target="_self" title="GRAPH_5:th.14">Th14</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">j</font> = <font color="Maroon" title="c13">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="c17">t1</font>, <font color="Maroon" title="c18">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="E15:96_1_1_1"/><span class="lab"><font color="Green" title="E86">A17</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E16:96_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j</font>
 <span class="kw">and </span><br/><a NAME="E17:96_1_1_1"/><span class="lab"><font color="Green" title="E87">A18</font></span>: 
<font color="Maroon" title="c15">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c18">t2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:96_1_1_1"><span class="lab"><font color="Green" title="E85">A16</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="c19">t1</font> = <font color="Maroon" title="c17">t1</font>, <font color="Maroon" title="c20">t2</font> = <font color="Maroon" title="c18">t2</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="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:96_1_1_1"><span class="lab"><font color="Green" title="E87">A18</font></span></a>, <a class="ref" href="graph_5.html#T14" target="_self" title="GRAPH_5:th.14">Th14</a></span>;<br/><span class="lab"><font color="Green" title="E88">A19</font></span>: <font color="Maroon" title="c7">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c15">t</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="#E5:96_1_1_1"><span class="lab"><font color="Green" title="E81">A12</font></span></a>, <a class="txt" href="#E6:96_1_1_1"><span class="lab"><font color="Green" title="E82">A13</font></span></a>, <a class="txt" href="#E9:96_1_1_1"><span class="lab"><font color="Green" title="E84">A15</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/>.= 
<font color="Maroon" title="c19">t1</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="#E15:96_1_1_1"><span class="lab"><font color="Green" title="E86">A17</font></span></a>, <a class="txt" href="#E17:96_1_1_1"><span class="lab"><font color="Green" title="E87">A18</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="E20:96_1_1_1"/><span class="lab"><font color="Green" title="E89">A20</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c20">t2</font>,<font color="Maroon" title="c2">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="#E1:96"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="ref" href="graph_5.html#T54" target="_self" title="GRAPH_5:th.54">Th54</a></span>;<br/><a NAME="E21:96_1_1_1"/><span class="lab"><font color="Green" title="E90">A21</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c19">t1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c20">t2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:96_1_1_1"><span class="lab"><font color="Green" title="E82">A13</font></span></a>, <a class="txt" href="#E17:96_1_1_1"><span class="lab"><font color="Green" title="E87">A18</font></span></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>;<br/><a NAME="E22:96_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="c2">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> <span class="p2">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c19">t1</font></span>)</span>,<font color="Maroon" title="c2">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="c20">t2</font>,<font color="Maroon" title="c2">W</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E70">A1</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="E23:96_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A22</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="c2">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c19">t1</font></span>)</span>,<font color="Maroon" title="c2">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:96_1_1_1"><span class="lab"><font color="Green" title="E89">A20</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c21">e</font> = <font color="Maroon" title="c7">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><a NAME="E24:96_1_1_1"/><span class="lab"><font color="Green" title="E92">A23</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_1_1_1"><span class="lab"><font color="Green" title="E79">A10</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c22">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E26:96_1_1_1"/><span class="lab"><font color="Green" title="E93">A24</font></span>: 
<font color="Maroon" title="c22">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="c7">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E27:96_1_1_1"/><span class="lab"><font color="Green" title="E94">A25</font></span>: 
not <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96_1_1_1"><span class="lab"><font color="Green" title="E80">A11</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E28:96_1_1_1"/><span class="lab"><font color="Green" title="E95">A26</font></span>: 
( <font color="Maroon" title="c22">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="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="c7">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> or <font color="Maroon" title="c22">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="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="c7">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:96_1_1_1"><span class="lab"><font color="Green" title="E93">A24</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E29:96_1_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="c19">t1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:96_1_1_1"><span class="lab"><font color="Green" title="E86">A17</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="c23">v3</font> = <font color="Maroon" title="c22">x</font> as   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:96_1_1_1"><span class="lab"><font color="Green" title="E92">A23</font></span></a>, <a class="txt" href="#E19:96_1_1_1"><span class="lab"><font color="Green" title="E88">A19</font></span></a>, <a class="txt" href="#E28:96_1_1_1"><span class="lab"><font color="Green" title="E95">A26</font></span></a>, <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="E31:96_1_1_1"/><span class="lab"><font color="Green" title="E96">A27</font></span>: 
<font color="Maroon" title="c5">v1</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="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="c7">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1"><span class="lab"><font color="Green" title="E77">A8</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</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="96_1_1_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="E1:96_1_1_1_2_1"/>
( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E97">A28</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <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="c4">P</font>,<font color="Maroon" title="c2">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="c2">W</font></span><br/><div class="add"><a NAME="E2:96_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A29</font></span>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">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="c19">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="c5">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1"><span class="lab"><font color="Green" title="E77">A8</font></span></a>, <a class="txt" href="#E19:96_1_1_1"><span class="lab"><font color="Green" title="E88">A19</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E3:96_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E99">A30</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="c19">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="c23">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="c1">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">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="c24">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="c19">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="c23">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:96_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E100">A31</font></span>: 
<font color="Maroon" title="c24">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="c19">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="c23">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="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">V</font></span><br/>

<a NAME="E2:96_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A32</font></span>: 
not <font color="Maroon" title="c24">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="c23">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="E3:96_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c24">x</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="c19">t1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A31</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="E4:96_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c24">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="c19">t1</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="#E15:96_1_1_1"><span class="lab"><font color="Green" title="E86">A17</font></span></a>, <a class="ref" href="graph_5.html#T29" target="_self" title="GRAPH_5:th.29">Th29</a></span>;<br/>
<a NAME="E5:96_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A33</font></span>: 
( <font color="Maroon" title="c24">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="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="c19">t1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> or <font color="Maroon" title="c24">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="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="c19">t1</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="E6:96_1_1_1_2_1_1_1"/>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">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="c19">t1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">v3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="txt" href="#E5:96"><span class="lab"><font color="Green" title="E74">A5</font></span></a>, <a class="txt" href="#E15:96_1_1_1"><span class="lab"><font color="Green" title="E86">A17</font></span></a>, <a class="txt" href="#E24:96_1_1_1"><span class="lab"><font color="Green" title="E92">A23</font></span></a>, <a class="txt" href="#E19:96_1_1_1"><span class="lab"><font color="Green" title="E88">A19</font></span></a>, <a class="txt" href="#E27:96_1_1_1"><span class="lab"><font color="Green" title="E94">A25</font></span></a>, <a class="txt" href="#E28:96_1_1_1"><span class="lab"><font color="Green" title="E95">A26</font></span></a>, <a class="txt" href="#E31:96_1_1_1"><span class="lab"><font color="Green" title="E96">A27</font></span></a>, <a class="txt" href="#E1:96_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E97">A28</font></span></a>, <a class="ref" href="graph_5.html#T35" target="_self" title="GRAPH_5:th.35">Th35</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="E7:96_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:96"><span class="lab"><font color="Green" title="E75">A6</font></span></a>, <a class="txt" href="#E15:96_1_1_1"><span class="lab"><font color="Green" title="E86">A17</font></span></a>, <a class="txt" href="#E2:96_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E98">A29</font></span></a>, <a class="txt" href="#E2:96_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A32</font></span></a>, <a class="txt" href="#E5:96_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E102">A33</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</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="E4:96_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E100">A34</font></span>: 
<font color="Maroon" title="c19">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:96_1_1_1"><span class="lab"><font color="Green" title="E86">A17</font></span></a></span>;<br/><a NAME="E5:96_1_1_1_2_1_1"/>
<font color="Maroon" title="c5">v1</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="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="c7">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1"><span class="lab"><font color="Green" title="E77">A8</font></span></a>, <a class="txt" href="#E24:96_1_1_1"><span class="lab"><font color="Green" title="E92">A23</font></span></a>, <a class="txt" href="#E1:96_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E97">A28</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E6:96_1_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">t1</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c23">v3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="txt" href="#E5:96"><span class="lab"><font color="Green" title="E74">A5</font></span></a>, <a class="txt" href="#E15:96_1_1_1"><span class="lab"><font color="Green" title="E86">A17</font></span></a>, <a class="txt" href="#E24:96_1_1_1"><span class="lab"><font color="Green" title="E92">A23</font></span></a>, <a class="txt" href="#E19:96_1_1_1"><span class="lab"><font color="Green" title="E88">A19</font></span></a>, <a class="txt" href="#E27:96_1_1_1"><span class="lab"><font color="Green" title="E94">A25</font></span></a>, <a class="txt" href="#E28:96_1_1_1"><span class="lab"><font color="Green" title="E95">A26</font></span></a>, <a class="txt" href="#E4:96_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E100">A34</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a>, <a class="ref" href="graph_5.html#T35" target="_self" title="GRAPH_5:th.35">Th35</a></span>;<br/><a NAME="E7:96_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A35</font></span>: 
<font color="Maroon" title="c19">t1</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c23">v3</font>,<font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E99">A30</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">then </span><span class="kw">consider </span><font color="Maroon" title="c24">q</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="c3">G</font><span class="kw"> such that </span><br/><a NAME="E9:96_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E102">A36</font></span>: 
<font color="Maroon" title="c24">q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c23">v3</font>,<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="ref" href="graph_5.html#T66" target="_self" title="GRAPH_5:th.66">Th66</a></span>;<br/><a NAME="E10:96_1_1_1_2_1_1"/>
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E5:96_1_1_1"><span class="lab"><font color="Green" title="E81">A12</font></span></a>, <a class="txt" href="#E1:96_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E97">A28</font></span></a></span>;<br/><a NAME="E11:96_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A37</font></span>: 
<font color="Maroon" title="c14">s</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c19">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">t1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>;<br/><a NAME="E12:96_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E104">A38</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c24">q</font>,<font color="Maroon" title="c2">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c19">t1</font>,<font color="Maroon" title="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:96_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E101">A35</font></span></a>, <a class="txt" href="#E9:96_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E102">A36</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>;<br/><a NAME="E13:96_1_1_1_2_1_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">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">q</font>,<font color="Maroon" title="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="txt" href="#E27:96_1_1_1"><span class="lab"><font color="Green" title="E94">A25</font></span></a>, <a class="txt" href="#E9:96_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E102">A36</font></span></a></span>;<br/><a NAME="E14:96_1_1_1_2_1_1"/><span class="kw">then </span>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c19">t1</font>,<font color="Maroon" title="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:96_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E104">A38</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="E15:96_1_1_1_2_1_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">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="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:96_1_1_1"><span class="lab"><font color="Green" title="E91">A22</font></span></a>, <a class="txt" href="#E11:96_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E103">A37</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E97">A39</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <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="c4">P</font>,<font color="Maroon" title="c2">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="c2">W</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c24">u</font> = <font color="Maroon" title="c14">s</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c19">t1</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="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:96_1_1_1"><span class="lab"><font color="Green" title="E90">A21</font></span></a>, <a class="ref" href="graph_5.html#T14" target="_self" title="GRAPH_5:th.14">Th14</a></span>;<br/><a NAME="E3:96_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E98">A40</font></span>: 
<font color="Maroon" title="c10">i</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="c7">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_1_1_1"><span class="lab"><font color="Green" title="E79">A10</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:96_1_1_1_2_1_2"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E97">A39</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E5:96_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A41</font></span>: 
<font color="Maroon" title="c10">i</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="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:96_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A42</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_1_1_1"><span class="lab"><font color="Green" title="E81">A12</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><div><span class="lab"><font color="Green" title="E101">A43</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_1_2_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:96_1_1_1_2_1_2_1"/>
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">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="c7">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <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> <font color="Maroon" title="c23">v3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:96_1_1_1_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A44</font></span>: 
<font color="Maroon" title="c23">v3</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="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="c7">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E99">A41</font></span></a>, <a class="txt" href="#E3:96_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E98">A40</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="E3:96_1_1_1_2_1_2_1"/>
<font color="Maroon" title="c7">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <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> <font color="Maroon" title="c10">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_1_1_1"><span class="lab"><font color="Green" title="E81">A12</font></span></a>, <a class="txt" href="#E6:96_1_1_1"><span class="lab"><font color="Green" title="E82">A13</font></span></a>, <a class="txt" href="#E5:96_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E99">A41</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="E4:96_1_1_1_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">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="c14">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:96_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E100">A42</font></span></a>, <a class="txt" href="#E2:96_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E102">A44</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">hence </span><a NAME="E5:96_1_1_1_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:96_1_1_1"><span class="lab"><font color="Green" title="E83">A14</font></span></a>, <a class="txt" href="#E27:96_1_1_1"><span class="lab"><font color="Green" title="E94">A25</font></span></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="E8:96_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E102">A45</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="c24">u</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="c23">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="c1">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_1_2_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c25">V3</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c26">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="c26">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="c24">u</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="c23">v3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:96_1_1_1_2_1_2_2"/><span class="lab"><font color="Green" title="E103">A46</font></span>: 
<font color="Maroon" title="c26">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="c24">u</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="c23">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="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">V</font></span><br/>

<a NAME="E2:96_1_1_1_2_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A47</font></span>: 
<font color="Maroon" title="c26">x</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">u</font>
 
<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="E3:96_1_1_1_2_1_2_2"/>
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c24">u</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="c14">s</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"><span class="p2">(<span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_1_1_1"><span class="lab"><font color="Green" title="E81">A12</font></span></a>, <a class="txt" href="#E15:96_1_1_1"><span class="lab"><font color="Green" title="E86">A17</font></span></a>, <a class="txt" href="#E5:96_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E99">A41</font></span></a>, <a class="ref" href="graph_5.html#T31" target="_self" title="GRAPH_5:th.31">Th31</a></span>;<br/>
<a NAME="E4:96_1_1_1_2_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A48</font></span>: 
( <font color="Maroon" title="c26">x</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="c14">s</font> or <font color="Maroon" title="c26">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"><span class="p2">(<span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96_1_1_1_2_1_2_2"><span class="lab"><font color="Green" title="E104">A47</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="E5:96_1_1_1_2_1_2_2"/>
not <font color="Maroon" title="c26">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="c23">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="#E1:96_1_1_1_2_1_2_2"><span class="lab"><font color="Green" title="E103">A46</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:96_1_1_1_2_1_2_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_1_1_1"><span class="lab"><font color="Green" title="E79">A10</font></span></a>, <a class="txt" href="#E7:96_1_1_1"><span class="lab"><font color="Green" title="E83">A14</font></span></a>, <a class="txt" href="#E19:96_1_1_1"><span class="lab"><font color="Green" title="E88">A19</font></span></a>, <a class="txt" href="#E28:96_1_1_1"><span class="lab"><font color="Green" title="E95">A26</font></span></a>, <a class="txt" href="#E7:96_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E101">A43</font></span></a>, <a class="txt" href="#E4:96_1_1_1_2_1_2_2"><span class="lab"><font color="Green" title="E105">A48</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</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="E9:96_1_1_1_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">u</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="c14">s</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/><a NAME="E10:96_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A49</font></span>: 
( <font color="Maroon" title="c24">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c24">u</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="c24">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">t1</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="#E15:96_1_1_1"><span class="lab"><font color="Green" title="E86">A17</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/><font color="Maroon" title="c24">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<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="#E5:96_1_1_1"><span class="lab"><font color="Green" title="E81">A12</font></span></a>, <a class="txt" href="#E5:96_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E99">A41</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/>.= 
<font color="Maroon" title="c7">r</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="#E5:96_1_1_1"><span class="lab"><font color="Green" title="E81">A12</font></span></a>, <a class="txt" href="#E6:96_1_1_1"><span class="lab"><font color="Green" title="E82">A13</font></span></a>, <a class="txt" href="#E5:96_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E99">A41</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="E12:96_1_1_1_2_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="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">u</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="c5">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1"><span class="lab"><font color="Green" title="E77">A8</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E13:96_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">u</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c23">v3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:96_1_1_1"><span class="lab"><font color="Green" title="E92">A23</font></span></a>, <a class="txt" href="#E19:96_1_1_1"><span class="lab"><font color="Green" title="E88">A19</font></span></a>, <a class="txt" href="#E28:96_1_1_1"><span class="lab"><font color="Green" title="E95">A26</font></span></a>, <a class="txt" href="#E7:96_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E101">A43</font></span></a>, <a class="txt" href="#E10:96_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E103">A49</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E14:96_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A50</font></span>: 
<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="c5">v1</font>,<font color="Maroon" title="c23">v3</font>,<font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:96_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E102">A45</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">then </span><span class="kw">consider </span><font color="Maroon" title="c25">q</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="c3">G</font><span class="kw"> such that </span><br/><a NAME="E16:96_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E105">A51</font></span>: 
<font color="Maroon" title="c25">q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c23">v3</font>,<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="ref" href="graph_5.html#T66" target="_self" title="GRAPH_5:th.66">Th66</a></span>;<br/><a NAME="E17:96_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E106">A52</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c25">q</font>,<font color="Maroon" title="c2">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="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:96_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E104">A50</font></span></a>, <a class="txt" href="#E16:96_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E105">A51</font></span></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></span>;<br/><a NAME="E18:96_1_1_1_2_1_2"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">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="c25">q</font>,<font color="Maroon" title="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="txt" href="#E27:96_1_1_1"><span class="lab"><font color="Green" title="E94">A25</font></span></a>, <a class="txt" href="#E16:96_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E105">A51</font></span></a></span>;<br/><a NAME="E19:96_1_1_1_2_1_2"/><span class="kw">then </span>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">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="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:96_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E106">A52</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="E20:96_1_1_1_2_1_2"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">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="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:96_1_1_1"><span class="lab"><font color="Green" title="E91">A22</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></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="96_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_2"/>
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c7">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</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="c4">P</font>,<font color="Maroon" title="c2">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="c2">W</font></span><br/><div class="add"><a NAME="E2:96_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="c7">r</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="c6">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="c1">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="c6">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="E3:96_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="c7">r</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="c6">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="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E4:96_1_1_2"/><span class="kw">then </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="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1"><span class="lab"><font color="Green" title="E77">A8</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="E5:96_1_1_2"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">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="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96"><span class="lab"><font color="Green" title="E71">A2</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>
<a NAME="E8:96"/>
<font color="Maroon" title="c4">P</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96"><span class="lab"><font color="Green" title="E74">A5</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="E9:96"/>
<font color="Maroon" title="c4">P</font> <a href="graph_5.html#R7" title="GRAPH_5:pred.7">is_shortestpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:96"><span class="lab"><font color="Green" title="E76">A7</font></span></a>, <a class="ref" href="graph_5.html#D17" target="_self" title="GRAPH_5:def.17">Def17</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>
