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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">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">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="c1">G</font><br/> for <font color="Olive" title="b2">W</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b2">W</font> <a href="graph_5.html#R6" title="GRAPH_5:pred.6">is_weight_of</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">pe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b1">pe</font>,<font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">pe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">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="c1">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_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">W</font> <a href="graph_5.html#R6" title="GRAPH_5:pred.6">is_weight_of</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">pe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c2">pe</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">W</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">W</font> <a href="graph_5.html#R6" title="GRAPH_5:pred.6">is_weight_of</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">pe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c2">pe</font>,<font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )</span><br/>





<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E5">A1</font></i>: 
( <font color="Maroon" title="c3">W</font> <a href="graph_5.html#R6" title="GRAPH_5:pred.6">is_weight_of</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">pe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c2">pe</font>,<font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c4">f</font> =  <a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c2">pe</font>,<font color="Maroon" title="c3">W</font>;<br/>
<a NAME="E2:4"/><i><font color="Green" title="E6">A2</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c2">pe</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">pe</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">pe</font> holds <br/><span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c2">pe</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="#E1:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="graph_5.html#D15" title="GRAPH_5:def.15">GRAPH_5:def 15</a></i>;<br/>
<a NAME="E3:4"/><i><font color="Green" title="E7">A3</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">pe</font>
 
<b>by </b><i><a class="txt" href="#E1:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c2">pe</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="#E1:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E2:4"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E5:4"/><b>then </b><i><font color="Green" title="E8">A4</font></i>: 
 <a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c2">pe</font>,<font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c2">pe</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<b>thus </b> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c2">pe</font>,<font color="Maroon" title="c3">W</font> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c2">pe</font>,<font color="Maroon" title="c3">W</font></span>)</span>
<b>by </b><i><a class="ref" href="graph_5.html#D16" title="GRAPH_5:def.16">GRAPH_5:def 16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c2">pe</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="#E5:4"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></i>
<br/>.= 
<font color="Maroon" title="c3">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E1:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E3:4"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="graph_5.html#D15" title="GRAPH_5:def.15">GRAPH_5:def 15</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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