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<b>let </b><font color="Maroon" title="c1">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>;<br/><b>let </b><font color="Maroon" title="c2">v1</font>, <font color="Maroon" title="c3">v2</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c4">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="c1">G</font>, <a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a> ;<br/>







<b>set </b><font color="Maroon" title="c5">EG</font> = the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:41_1"/><i><font color="Green" title="E34">A1</font></i>: 
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 ;<br/><b>assume </b><a NAME="E2:41_1"/>
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> or  not <font color="Olive" title="b1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font> )
 ;<br/><a NAME="E3:41_1"/><b>then </b>
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="ref" href="graphsp.html#D7" target="_self" title="GRAPHSP:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E4:41_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E1:41_1"><i><font color="Green" title="E34">A1</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:41"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> &amp; <font color="Olive" title="b1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:41"/><i><font color="Green" title="E34">A2</font></i>: 
(  <a href="graphsp.html#K10" title="GRAPHSP:func.10">Edge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e</font> &amp; <font color="Maroon" title="c6">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="c1">G</font> &amp; <font color="Maroon" title="c6">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">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="E5:41"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">W</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E4:41"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:41"/><b>then </b>
<font color="Maroon" title="c4">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a> 
 
<b>by </b><i><a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E8:41"/><i><font color="Green" title="E35">A3</font></i>: 
( <font color="Maroon" title="c4">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> &amp; <font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="graph_5.html#D12" title="GRAPH_5:def.12">GRAPH_5:def 12</a></i>;<br/>
<b>thus </b><a NAME="E9:41"/>
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="graphsp.html#E4:41"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="graphsp.html#E8:41"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="graphsp.html#D7" target="_self" title="GRAPHSP:def.7">Def7</a></i>;<br/>


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