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<b>let </b><font color="Maroon" title="c1">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font> being <a href="graph_1.html#V3" title="GRAPH_1:attr.3">oriented</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a><br/> for <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Olive" title="b1">G</font><br/> for <font color="Olive" title="b4">W</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Olive" title="b1">G</font>, <a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a>   st <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b3">v2</font> holds <br/> <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Olive" title="b4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">W</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">e</font></span><br/><b>let </b><font color="Maroon" title="c2">G</font> be  <a href="graph_1.html#V3" title="GRAPH_1:attr.3">oriented</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c2">G</font><br/> for <font color="Olive" title="b3">W</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>, <a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a>   st <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font> holds <br/> <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font>,<font color="Olive" title="b3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">W</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">e</font></span><br/><b>let </b><font color="Maroon" title="c3">v1</font>, <font color="Maroon" title="c4">v2</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>, <a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a>   st <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font> holds <br/> <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font>,<font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">e</font></span><br/><b>let </b><font color="Maroon" title="c5">W</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>, <a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font> implies  <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font>,<font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">e</font> )</span><br/>









<b>set </b><font color="Maroon" title="c6">EG</font> = the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>;<br/>
<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E35">A1</font></i>: 
( <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font>,<font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">e</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:42"/><i><font color="Green" title="E36">A2</font></i>: 
(  <a href="graphsp.html#K10" title="GRAPHSP:func.10">Edge</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e1</font> &amp; <font color="Maroon" title="c7">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c7">e1</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font> )
 <b>by </b><i><a class="ref" href="graphsp.html#D6" target="_self" title="GRAPHSP:def.6">Def6</a></i>;<br/>
<a NAME="E4:42"/>
<font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e1</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:42"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="graphsp.html#E3:42"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="graphsp.html#T10" target="_self" title="GRAPHSP:th.10">Th10</a></i>;<br/>
<b>hence </b><a NAME="E5:42"/>
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font>,<font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">e</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:42"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="graphsp.html#D7" target="_self" title="GRAPHSP:def.7">Def7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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