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<b>let </b><font color="Maroon" title="c1">W</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c3">qe</font>, <font color="Maroon" title="c4">pe</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>;<br/>







<b>assume </b><a NAME="E1:72"/><i><font color="Green" title="E56">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">qe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">qe</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">pe</font> &amp; <font color="Maroon" title="c1">W</font> <a href="graph_5.html#R5" title="GRAPH_5:pred.5">is_weight&gt;=0of</a> <font color="Maroon" title="c2">G</font> )
 ;<br/>

<a NAME="E2:72"/><b>then </b><i><font color="Green" title="E57">A2</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="c3">qe</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">f</font> =  <a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c4">pe</font>,<font color="Maroon" title="c1">W</font>;<br/>
<b>set </b><font color="Maroon" title="c6">g</font> =  <a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c3">qe</font>,<font color="Maroon" title="c1">W</font>;<br/>
<a NAME="E3:72"/><i><font color="Green" title="E58">A3</font></i>: 
<font color="Maroon" title="c1">W</font> <a href="graph_5.html#R6" title="GRAPH_5:pred.6">is_weight_of</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:72"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="graph_5.html#T50" target="_self" title="GRAPH_5:th.50">Th50</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">j</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> <b> such that </b><br/><a NAME="E5:72"/><i><font color="Green" title="E59">A4</font></i>: 
( <font color="Maroon" title="c7">j</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="c4">pe</font> &amp; <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c4">pe</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c3">qe</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 )
 <b>by </b><i><a class="txt" href="graph_5.html#E1:72"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="graph_5.html#E2:72"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="graph_5.html#T52" target="_self" title="GRAPH_5:th.52">Th52</a></i>;<br/>
<a NAME="E6:72"/><i><font color="Green" title="E60">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">pe</font> )
 
<b>by </b><i><a class="txt" href="graph_5.html#E5:72"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:72"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">pe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c4">pe</font>,<font color="Maroon" title="c1">W</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_5.html#E3:72"><i><font color="Green" title="E58">A3</font></i></a>, <a class="ref" href="graph_5.html#D15" target="_self" title="GRAPH_5:def.15">Def15</a></i>;<br/>
<a NAME="E8:72"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">pe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c4">pe</font>,<font color="Maroon" title="c1">W</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:72"/><i><font color="Green" title="E61">A6</font></i>: 
 <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="c4">pe</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">m</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:72"><i><font color="Green" title="E60">A5</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">m</font> = <font color="Maroon" title="c8">m</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">f1</font>, <font color="Maroon" title="c11">f2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E13:72"/><i><font color="Green" title="E62">A7</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">m</font> &amp;  <a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c4">pe</font>,<font color="Maroon" title="c1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">f2</font> )
 <b>by </b><i><a class="txt" href="graph_5.html#E10:72"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="finseq_2.html#T26" title="FINSEQ_2:th.26">FINSEQ_2:26</a></i>;<br/>
<a NAME="E14:72"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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> <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c4">pe</font>,<font color="Maroon" title="c1">W</font></span>)</span> 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="c4">pe</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:72"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="graph_5.html#T51" target="_self" title="GRAPH_5:th.51">Th51</a></i>;<br/>
<a NAME="E15:72"/><b>then </b>
for <font color="Olive" title="b1">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c4">pe</font>,<font color="Maroon" title="c1">W</font></span>)</span> holds <br/><font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="graph_5.html#E71"><i><font color="Green" title="E54">Lm3</font></i></a></i>;<br/>
<a NAME="E16:72"/><b>then </b><i><font color="Green" title="E63">A8</font></i>: 
( ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="c10">f1</font> holds <br/><font color="Maroon" title="c10">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="c11">f2</font> holds <br/><font color="Maroon" title="c11">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 
<b>by </b><i><a class="txt" href="graph_5.html#E13:72"><i><font color="Green" title="E62">A7</font></i></a>, <a class="txt" href="graph_5.html#E72"><i><font color="Green" title="E55">Lm4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E18:72"/><i><font color="Green" title="E64">A9</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">n</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:72"><i><font color="Green" title="E60">A5</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<a NAME="E19:72"/>
 <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="c3">qe</font>,<font color="Maroon" title="c1">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="c3">qe</font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E3:72"><i><font color="Green" title="E58">A3</font></i></a>, <a class="ref" href="graph_5.html#D15" target="_self" title="GRAPH_5:def.15">Def15</a></i>;<br/>
<a NAME="E20:72"/><b>then </b>
 <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="c3">qe</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">qe</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E21:72"/><b>then </b>
 <a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c3">qe</font>,<font color="Maroon" title="c1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&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="c3">qe</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1</span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:72"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E22:72"/><b>then </b><i><font color="Green" title="E65">A10</font></i>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c3">qe</font>,<font color="Maroon" title="c1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c3">qe</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1
 
<b>by </b><i><a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <font color="Maroon" title="c14">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E24:72"/><i><font color="Green" title="E66">A11</font></i>: 
<font color="Maroon" title="c10">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c14">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="graph_5.html#E13:72"><i><font color="Green" title="E62">A7</font></i></a>, <a class="txt" href="graph_5.html#E18:72"><i><font color="Green" title="E64">A9</font></i></a>, <a class="ref" href="finseq_2.html#T22" title="FINSEQ_2:th.22">FINSEQ_2:22</a></i>;<br/>
<a NAME="E25:72"/><i><font color="Green" title="E67">A12</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">h</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="graph_5.html#E13:72"><i><font color="Green" title="E62">A7</font></i></a>, <a class="txt" href="graph_5.html#E24:72"><i><font color="Green" title="E66">A11</font></i></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/>
<a NAME="E26:72"/>
<font color="Maroon" title="c7">j</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="c10">f1</font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E6:72"><i><font color="Green" title="E60">A5</font></i></a>, <a class="txt" href="graph_5.html#E13:72"><i><font color="Green" title="E62">A7</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E27:72"/><b>then </b>
<font color="Maroon" title="c10">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c3">qe</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1
 
<b>by </b><i><a class="txt" href="graph_5.html#E5:72"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="graph_5.html#E13:72"><i><font color="Green" title="E62">A7</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/>
<a NAME="E28:72"/><b>then </b><i><font color="Green" title="E68">A13</font></i>: 
<font color="Maroon" title="c14">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c3">qe</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1
 
<b>by </b><i><a class="txt" href="graph_5.html#E24:72"><i><font color="Green" title="E66">A11</font></i></a>, <a class="txt" href="graph_5.html#E25:72"><i><font color="Green" title="E67">A12</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E29:72"/><i><font color="Green" title="E69">A14</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c11">f2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="graph_5.html#E16:72"><i><font color="Green" title="E63">A8</font></i></a>, <a class="ref" href="rvsum_1.html#T114" title="RVSUM_1:th.114">RVSUM_1:114</a></i>;<br/>
<a NAME="E30:72"/><i><font color="Green" title="E70">A15</font></i>: 
 <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="c4">pe</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c10">f1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c11">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_5.html#E13:72"><i><font color="Green" title="E62">A7</font></i></a>, <a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>;<br/>
<a NAME="E31:72"/>
for <font color="Olive" title="b1">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c10">f1</font> holds <br/><font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="graph_5.html#E16:72"><i><font color="Green" title="E63">A8</font></i></a>, <a class="txt" href="graph_5.html#E71"><i><font color="Green" title="E54">Lm3</font></i></a></i>;<br/>
<a NAME="E32:72"/><b>then </b>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="c13">h</font> holds <br/><font color="Maroon" title="c13">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="graph_5.html#E24:72"><i><font color="Green" title="E66">A11</font></i></a>, <a class="txt" href="graph_5.html#E72"><i><font color="Green" title="E55">Lm4</font></i></a></i>;<br/>
<a NAME="E33:72"/><b>then </b><i><font color="Green" title="E71">A16</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c13">h</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T114" title="RVSUM_1:th.114">RVSUM_1:114</a></i>;<br/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c10">f1</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c13">h</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c14">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="graph_5.html#E24:72"><i><font color="Green" title="E66">A11</font></i></a>, <a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c13">h</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c3">qe</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graph_5.html#E28:72"><i><font color="Green" title="E68">A13</font></i></a>, <a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></i>
;<br/>
<a NAME="E35:72"/><b>then </b>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c10">f1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> <font color="Maroon" title="c3">qe</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="graph_5.html#E33:72"><i><font color="Green" title="E71">A16</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E36:72"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c3">qe</font>,<font color="Maroon" title="c1">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c4">pe</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E22:72"><i><font color="Green" title="E65">A10</font></i></a>, <a class="txt" href="graph_5.html#E29:72"><i><font color="Green" title="E69">A14</font></i></a>, <a class="txt" href="graph_5.html#E30:72"><i><font color="Green" title="E70">A15</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>


</div>
