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<span class="kw">let </span><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:()"><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#NM3" title="GRAPH_1:NM.3">Graph</a><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="Olive" title="b1">G</font><br/> for <font color="Olive" title="b4">D</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Olive" title="b1">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>  st <font color="Olive" title="b4">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K4" title="GRAPH_5:func.4">AcyclicPaths</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b3">v2</font> holds <br/> ex <font color="Olive" title="b5">pe</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="Olive" title="b1">G</font> st <br/>( <font color="Olive" title="b5">pe</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">D</font> &amp; ( for <font color="Olive" title="b6">qe</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="Olive" title="b1">G</font>  st <font color="Olive" title="b6">qe</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">D</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b5">pe</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" title="b6">qe</font>,<font color="Maroon" title="c1">W</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">G</font> be   <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">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="c2">G</font><br/> for <font color="Olive" title="b3">D</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>  st <font color="Olive" title="b3">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K4" title="GRAPH_5:func.4">AcyclicPaths</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font> holds <br/> ex <font color="Olive" title="b4">pe</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="c2">G</font> st <br/>( <font color="Olive" title="b4">pe</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">D</font> &amp; ( for <font color="Olive" title="b5">qe</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="c2">G</font>  st <font color="Olive" title="b5">qe</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">D</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b4">pe</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" title="b5">qe</font>,<font color="Maroon" title="c1">W</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">v1</font>, <font color="Maroon" title="c4">v2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">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">D</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>  st <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K4" title="GRAPH_5:func.4">AcyclicPaths</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font> holds <br/> ex <font color="Olive" title="b2">pe</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="c2">G</font> st <br/>( <font color="Olive" title="b2">pe</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> &amp; ( for <font color="Olive" title="b3">qe</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="c2">G</font>  st <font color="Olive" title="b3">qe</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b2">pe</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" title="b3">qe</font>,<font color="Maroon" title="c1">W</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">D</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</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="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K4" title="GRAPH_5:func.4">AcyclicPaths</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font> implies  ex <font color="Olive" title="b1">pe</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="c2">G</font> st <br/>( <font color="Olive" title="b1">pe</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; ( for <font color="Olive" title="b2">qe</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="c2">G</font>  st <font color="Olive" title="b2">qe</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b1">pe</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" title="b2">qe</font>,<font color="Maroon" title="c1">W</font> ) ) )</span><br/>









<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="graph_5.html#M1" title="GRAPH_5:mode.1">Element</a> of <font color="Maroon" title="c5">D</font>) <span class="kw">-&gt; </span>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> =  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> $1,<font color="Maroon" title="c1">W</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="graph_5.html#M1" title="GRAPH_5:mode.1">Element</a> of <font color="Maroon" title="c5">D</font><span class="kw"> such that </span><br/><a NAME="E2:77"/><span class="lab"><font color="Green" title="E57">A1</font></span>: 
for <font color="Olive" title="b1">y</font> being   <a href="graph_5.html#M1" title="GRAPH_5:mode.1">Element</a> of <font color="Maroon" title="c5">D</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">x</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="graph_5.html#S2" target="_self" title="GRAPH_5:sch.2">GRAPH_5:sch 2</a>();<br/></span>
<span class="kw">assume </span><a NAME="E3:77"/>
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K4" title="GRAPH_5:func.4">AcyclicPaths</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">pe</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="c2">G</font> st <br/>( <font color="Olive" title="b1">pe</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; ( for <font color="Olive" title="b2">qe</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="c2">G</font>  st <font color="Olive" title="b2">qe</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b1">pe</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" title="b2">qe</font>,<font color="Maroon" title="c1">W</font> ) )</span><br/>

<a NAME="E4:77"/><span class="kw">then </span>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K4" title="GRAPH_5:func.4">AcyclicPaths</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">p</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="c2">G</font><span class="kw"> such that </span><br/><a NAME="E6:77"/><span class="lab"><font color="Green" title="E58">A2</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">and </span><br/><a NAME="E7:77"/>
<font color="Maroon" title="c7">p</font> <a href="graph_5.html#R3" title="GRAPH_5:pred.3">is_acyclicpath_of</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font>
 ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">p</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">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; ( for <font color="Olive" title="b1">qe</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="c2">G</font>  st <font color="Olive" title="b1">qe</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">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="Olive" title="b1">qe</font>,<font color="Maroon" title="c1">W</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E8:77"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:77"><span class="lab"><font color="Green" title="E58">A2</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"> for <font color="Olive" title="b1">qe</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="c2">G</font>  st <font color="Olive" title="b1">qe</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">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="Olive" title="b1">qe</font>,<font color="Maroon" title="c1">W</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">pe</font> be    <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="c2">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="c8">pe</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> implies  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">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="c8">pe</font>,<font color="Maroon" title="c1">W</font> )</span><br/>

<span class="kw">assume </span><a NAME="E9:77"/>
<font color="Maroon" title="c8">pe</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</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="c7">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="c8">pe</font>,<font color="Maroon" title="c1">W</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c8">pe</font> as    <a href="graph_5.html#M1" title="GRAPH_5:mode.1">Element</a> of <font color="Maroon" title="c5">D</font> ;<br/>
<a NAME="E11:77"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">x</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">y</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77"><span class="lab"><font color="Green" title="E57">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:77"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">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="c8">pe</font>,<font color="Maroon" title="c1">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:77"><span class="lab"><font color="Green" title="E58">A2</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>
