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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">qe</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">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>assume </b><a NAME="E1:73"/><i><font color="Green">E39</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">qe</font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">qe</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">pe</font> &amp; <font color="Maroon">W</font> <a href="graph_5.html#R5" target="_self">is_weight&gt;=0of</a> <font color="Maroon">G</font> )
 ;<br/>

<a NAME="E2:73"/><b>then </b><i><font color="Green">E45</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">qe</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font>;<br/>
<b>set </b><font color="Maroon">g</font> =  <a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font>;<br/>
<a NAME="E3:73"/><i><font color="Green">E47</font></i>: 
<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><a class="ref" href="graph_5.html#T6" target="_self">Th6</a>, <a class="txt" href="graph_5.html#E16:73"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:73"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">pe</font> &amp; <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 )
 <b>by </b><i><a class="ref" href="graph_5.html#T6" target="_self">Th6</a>, <a class="ref" href="graph_5.html#T7" target="_self">Th7</a>, <a class="ref" href="graph_5.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E6:73"/><i><font color="Green">E51</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">pe</font> )
 
<b>by </b><i><a class="txt" href="graph_5.html#E18:73"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:73"/>
 <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> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="graph_5.html#E13:73"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E8:73"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">pe</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:73"/><i><font color="Green">E52</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E21:73"><i><font color="Green">E63</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon">f1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">f2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E13:73"/><i><font color="Green">E53</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  <a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f2</font> )
 <b>by </b><i><a class="txt" href="graph_5.html#E22:73"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_2.html#T26">FINSEQ_2:26</a></i>;<br/>
<a NAME="E14:73"/>
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> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="graph_5.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E15:73"/><b>then </b>
for <font color="Olive">y</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span> holds <br/><font color="Olive">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="graph_5.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E16:73"/><b>then </b><i><font color="Green">E54</font></i>: 
( ( 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">f1</font> holds <br/><font color="Maroon">f1</font> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) &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">f2</font> holds <br/><font color="Maroon">f2</font> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) )
 
<b>by </b><i><a class="txt" href="graph_5.html#E24:73"><i><font color="Green">E65</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E18:73"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E21:73"><i><font color="Green">E63</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E19:73"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">qe</font>
 
<b>by </b><i>, <a class="txt" href="graph_5.html#E13:73"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E20:73"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">qe</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E21:73"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span> <a href="goboard1.html#K1">.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="graph_5.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E22:73"/><i><font color="Green">E64</font></i>: 
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span> <a href="goboard1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="graph_5.html#E29:73"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>;<br/>
<b>consider </b><font color="Maroon">h</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">d</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E24:73"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="graph_5.html#E24:73"><i><font color="Green">E65</font></i></a>, <a class="txt" href="graph_5.html#E28:73"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_2.html#T22">FINSEQ_2:22</a></i>;<br/>
<a NAME="E25:73"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="graph_5.html#E24:73"><i><font color="Green">E65</font></i></a>, , <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/>
<a NAME="E26:73"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E21:73"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_5.html#E24:73"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E27:73"/><b>then </b>
<font color="Maroon">f1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span> <a href="goboard1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="graph_5.html#E18:73"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_5.html#E24:73"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<a NAME="E28:73"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span> <a href="goboard1.html#K1">.</a> 1
 
<b>by </b><i>, <a class="ref" href="graph_5.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E29:73"/><i><font color="Green">E68</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">f2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="graph_5.html#E25:73"><i><font color="Green">E66</font></i></a>, <a class="ref" href="rvsum_1.html#T114">RVSUM_1:114</a></i>;<br/>
<a NAME="E30:73"/><i><font color="Green">E87</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">f1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_5.html#E24:73"><i><font color="Green">E65</font></i></a>, <a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>;<br/>
<a NAME="E31:73"/>
for <font color="Olive">y</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> holds <br/><font color="Olive">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="graph_5.html#E25:73"><i><font color="Green">E66</font></i></a>, <a class="ref" href="graph_5.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E32:73"/><b>then </b>
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">h</font> holds <br/><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E33:73"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">h</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T114">RVSUM_1:114</a></i>;<br/>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">f1</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">h</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">h</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span> <a href="goboard1.html#K1">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graph_5.html#E9"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>
;<br/>
<a NAME="E35:73"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span> <a href="goboard1.html#K1">.</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E36:73"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="txt" href="graph_5.html#E10"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="graph_5.html#T9" target="_self">Th9</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>


</div>
