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<b>let </b><font color="Maroon">W</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">pe</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>







<b>assume </b><a NAME="E1:69"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">W</font> <a href="graph_5.html#R5" target="_self">is_weight&gt;=0of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font> )
 ;<br/>

<a NAME="E2:69"/><b>then </b>
<font color="Maroon">W</font> <a href="graph_5.html#R6" target="_self">is_weight_of</a> <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:69"/><b>then </b><i><font color="Green">E45</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">pe</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">pe</font> holds <br/><font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="graph_5.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E4:69"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">W</font> is   <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" target="_self">Real&gt;=0</a> 
 
<b>by </b><i><a class="ref" href="graph_5.html#T4" target="_self">Th4</a>, <a class="txt" href="graph_5.html#E6:69"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E5:69"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E6:69"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="graph_5.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E7:69"/>
<font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="graph_5.html#T5" target="_self">Th5</a>, <a class="ref" href="graph_5.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E8:69"/><b>then </b>
<font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K8" target="_self">Real&gt;=0</a> 
 
<b>by </b><i>, <a class="ref" href="graph_5.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</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="E10:69"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</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 )
 ;<br/>
<b>thus </b><a NAME="E11:69"/>
<font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/>


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