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<b>let </b><font color="Maroon">G</font> be  <a href="graph_1.html#V3">oriented</a> <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">v2</font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">W</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>, <a href="graph_5.html#K8">Real&gt;=0</a> ;<br/>







<b>set </b><font color="Maroon">EG</font> = the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:43_1"/><i><font color="Green">E34</font></i>: 
 <a href="graphsp.html#K13" target="_self">Weight</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><b>assume </b><a NAME="E2:43_1"/>
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">e</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> or  not <font color="Olive">e</font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font> )
 ;<br/><a NAME="E3:43_1"/><b>then </b>
 <a href="graphsp.html#K13" target="_self">Weight</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="graphsp.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E4:43_1"/>
contradiction
 <b>by </b><i><a class="ref" href="graphsp.html#T2" target="_self">Th2</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:43"/>
 ex <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">e</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> &amp; <font color="Olive">e</font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:43"/><i><font color="Green">E52</font></i>: 
(  <a href="graphsp.html#K11" target="_self">Edge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">e</font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:43"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="graphsp.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:43"/><b>then </b>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K8">Real&gt;=0</a> 
 
<b>by </b><i><a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:43"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="graph_5.html#D12">GRAPH_5:def 12</a></i>;<br/>
<b>thus </b><a NAME="E9:43"/>
 <a href="graphsp.html#K13" target="_self">Weight</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="graphsp.html#T2" target="_self">Th2</a>, <a class="ref" href="graphsp.html#T3" target="_self">Th3</a>, <a class="ref" href="graphsp.html#T1" target="_self">Th1</a></i>;<br/>


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