<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">W</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a><br/>  for <font color="Olive" title="b2">pe</font>, <font color="Olive" title="b3">qe</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="Olive" title="b1">G</font>  st <font color="Maroon" title="c1">W</font> <a href="graph_5.html#R6" title="GRAPH_5:pred.6">is_weight_of</a> <font color="Olive" title="b1">G</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b3">qe</font></span>)</span>,<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#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b2">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b3">qe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">G</font> be   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">pe</font>, <font color="Olive" title="b2">qe</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="c2">G</font>  st <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> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">qe</font></span>)</span>,<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#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b1">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b2">qe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">pe</font>, <font color="Maroon" title="c4">qe</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="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>,<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#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c3">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c4">qe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">r</font> = <font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">f</font> =  <a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>,<font color="Maroon" title="c1">W</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">g</font> =  <a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<font color="Maroon" title="c3">pe</font>,<font color="Maroon" title="c1">W</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">h</font> =  <a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<font color="Maroon" title="c4">qe</font>,<font color="Maroon" title="c1">W</font>);<br/>
<span class="kw">assume </span><a NAME="E1:76"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>,<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#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c3">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c4">qe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span></span><br/>

<a NAME="E2:76"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A2</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">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="c3">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#D15" target="_self" title="GRAPH_5:def.15">Def15</a></span>;<br/>
<a NAME="E3:76"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">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="c3">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E4:76"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">qe</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">qe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="graph_5.html#D15" target="_self" title="GRAPH_5:def.15">Def15</a></span>;<br/>
<a NAME="E5:76"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">qe</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">qe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E6:76"/><span class="lab"><font color="Green" title="E64">A6</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</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> <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>,<font color="Maroon" title="c1">W</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="graph_5.html#D15" target="_self" title="GRAPH_5:def.15">Def15</a></span>;<br/>
<a NAME="E7:76"/><span class="kw">then </span>
 <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> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>,<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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E8:76"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A7</font></span>: 
 <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> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>,<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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<font color="Maroon" title="c3">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<font color="Maroon" title="c4">qe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:76"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E5:76"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E66">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">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">qe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<font color="Maroon" title="c3">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">i</font></span>)</span> <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="c4">qe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:76_1"/><span class="lab"><font color="Green" title="E67">A9</font></span>: 
<font color="Maroon" title="c9">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">qe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<font color="Maroon" title="c3">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">i</font></span>)</span> <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="c4">qe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span><br/><a NAME="E2:76_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E3:76_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A10</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<font color="Maroon" title="c3">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E4:76_1"/>
<font color="Maroon" title="c9">i</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> <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<font color="Maroon" title="c4">qe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76_1"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:76_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<font color="Maroon" title="c3">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">i</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> <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>,<font color="Maroon" title="c1">W</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:76"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E6:76_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A11</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<font color="Maroon" title="c3">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">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> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>,<font color="Maroon" title="c1">W</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:76_1"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E7:76_1"/>
( <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<font color="Maroon" title="c4">qe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">qe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<font color="Maroon" title="c3">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">qe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E4:76"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="txt" href="#E3:76"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E1:76_1"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="ref" href="graph_5.html#D15" target="_self" title="GRAPH_5:def.15">Def15</a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E8:76_1"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<font color="Maroon" title="c3">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">i</font></span>)</span> <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="c4">qe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E6:76"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="txt" href="#E6:76_1"><span class="lab"><font color="Green" title="E69">A11</font></span></a>, <a class="ref" href="graph_5.html#D15" target="_self" title="GRAPH_5:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">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="c3">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</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">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:76_2"/><span class="lab"><font color="Green" title="E67">A12</font></span>: 
<font color="Maroon" title="c9">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="c3">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</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">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span><br/><a NAME="E2:76_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">i</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> <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<font color="Maroon" title="c3">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E3:76_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A13</font></span>: 
<font color="Maroon" title="c9">i</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> <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>,<font color="Maroon" title="c1">W</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:76"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E4:76_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76_2"><span class="lab"><font color="Green" title="E67">A12</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:76_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A14</font></span>: 
<font color="Maroon" title="c9">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> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>,<font color="Maroon" title="c1">W</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:76_2"><span class="lab"><font color="Green" title="E68">A13</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E6:76_2"/>
( <span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<font color="Maroon" title="c3">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">pe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E2:76"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E1:76_2"><span class="lab"><font color="Green" title="E67">A12</font></span></a>, <a class="ref" href="graph_5.html#D15" target="_self" title="GRAPH_5:def.15">Def15</a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E7:76_2"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</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">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E6:76"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="txt" href="#E5:76_2"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="ref" href="graph_5.html#D15" target="_self" title="GRAPH_5:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E11:76"/><span class="kw">then </span>
 <a href="graph_5.html#K9" title="GRAPH_5:func.9">RealSequence</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>,<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">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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">qe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:76"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="txt" href="#E9:76"><span class="lab"><font color="Green" title="E66">A8</font></span></a>, <a class="ref" href="finseq_3.html#T38" title="FINSEQ_3:th.38">FINSEQ_3:38</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:76"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">qe</font></span>)</span>,<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#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c3">pe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c4">qe</font>,<font color="Maroon" title="c1">W</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T75" title="RVSUM_1:th.75">RVSUM_1:75</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
