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<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">qe</font>, <font color="Olive" title="b3">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="Olive" title="b1">G</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">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="Olive" title="b2">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="Olive" title="b3">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="Olive" title="b1">G</font> holds <br/> <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>) <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="Olive" title="b3">pe</font>,<font color="Maroon" title="c1">W</font>)</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">qe</font>, <font color="Olive" title="b2">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="c2">G</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b1">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="Olive" title="b2">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> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b1">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="Olive" title="b2">pe</font>,<font color="Maroon" title="c1">W</font>)</span><br/><span class="kw">let </span><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="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"> (  <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> implies  <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>) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:71"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
 <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
 <span class="kw">and </span><br/><a NAME="E2:71"/><span class="lab"><font color="Green" title="E55">A2</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E3:71"/><span class="lab"><font color="Green" title="E56">A3</font></span>: 
<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>
 ; <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> (<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>)</span><br/>

<span class="kw">set </span><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/>
<span class="kw">set </span><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="E4:71"/>
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">j</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E6:71"/><span class="lab"><font color="Green" title="E57">A4</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:71"/><span class="lab"><font color="Green" title="E58">A5</font></span>: 
<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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E3:71"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="ref" href="graph_5.html#T46" target="_self" title="GRAPH_5:th.46">Th50</a>, <a class="ref" href="graph_5.html#T48" target="_self" title="GRAPH_5:th.48">Th52</a></span>;<br/>
<a NAME="E8:71"/><span class="lab"><font color="Green" title="E59">A6</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="ref" href="graph_5.html#T46" target="_self" title="GRAPH_5:th.46">Th50</a></span>;<br/>
<a NAME="E9:71"/><span class="kw">then </span>
 <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>
 
<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="E10:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A7</font></span>: 
 <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>
 
<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="E11:71"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="ref" href="graph_5.html#D15" target="_self" title="GRAPH_5:def.15">Def15</a></span>;<br/>
<a NAME="E12:71"/><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> (<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>
 
<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="E13:71"/><span class="kw">then </span>
 <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="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="c3">qe</font>,<font color="Maroon" title="c1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/>
<a NAME="E14:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A8</font></span>: 
 <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#K1" title="SEQ_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T11" title="FINSOP_1:th.11">FINSOP_1:11</a></span>;<br/>
<a NAME="E15:71"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:71"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E17:71"/><span class="lab"><font color="Green" title="E62">A9</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> (<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:71"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">m</font> = <font color="Maroon" title="c8">m</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E20:71"/><span class="lab"><font color="Green" title="E63">A10</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E21:71"/>
 <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>
 <span class="kw">and </span><br/><a NAME="E22:71"/><span class="lab"><font color="Green" title="E64">A11</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:71"><span class="lab"><font color="Green" title="E62">A9</font></span></a>, <a class="ref" href="finseq_2.html#T23" title="FINSEQ_2:th.23">FINSEQ_2:23</a></span>;<br/>
<a NAME="E23:71"/><span class="lab"><font color="Green" title="E65">A12</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:71"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">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="c13">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E25:71"/><span class="lab"><font color="Green" title="E66">A13</font></span>: 
<font color="Maroon" title="c10">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c13">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:71"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></span>;<br/>
<a NAME="E26:71"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:71"><span class="lab"><font color="Green" title="E65">A12</font></span></a>, <a class="txt" href="#E20:71"><span class="lab"><font color="Green" title="E63">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="E27:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A14</font></span>: 
<font color="Maroon" title="c10">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:71"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="txt" href="#E22:71"><span class="lab"><font color="Green" title="E64">A11</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/>
<a NAME="E28:71"/>
<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="c12">h</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:71"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="txt" href="#E25:71"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="ref" href="finseq_2.html#T16" title="FINSEQ_2:th.16">FINSEQ_2:16</a></span>;<br/>
<a NAME="E29:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A15</font></span>: 
<font color="Maroon" title="c13">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#K1" title="SEQ_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:71"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="txt" href="#E27:71"><span class="lab"><font color="Green" title="E67">A14</font></span></a>, <a class="ref" href="finseq_1.html#T42" title="FINSEQ_1:th.42">FINSEQ_1:42</a></span>;<br/>
<a NAME="E30:71"/>
 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#K1" title="SEQ_1:func.1">.</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="ref" href="graph_5.html#T47" target="_self" title="GRAPH_5:th.47">Th51</a></span>;<br/>
<a NAME="E31:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A16</font></span>: 
 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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="graph_5.html#E67"><span class="lab"><font color="Green" title="E52">Lm3</font></span></a></span>;<br/>
<a NAME="E32:71"/><span class="kw">then </span>
 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#K1" title="SEQ_1:func.1">.</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:71"><span class="lab"><font color="Green" title="E64">A11</font></span></a>, <a class="txt" href="graph_5.html#E68"><span class="lab"><font color="Green" title="E53">Lm4</font></span></a></span>;<br/>
<a NAME="E33:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A17</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T84" title="RVSUM_1:th.84">RVSUM_1:84</a></span>;<br/>
<a NAME="E34:71"/>
 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#K1" title="SEQ_1:func.1">.</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:71"><span class="lab"><font color="Green" title="E64">A11</font></span></a>, <a class="txt" href="#E31:71"><span class="lab"><font color="Green" title="E69">A16</font></span></a>, <a class="txt" href="graph_5.html#E68"><span class="lab"><font color="Green" title="E53">Lm4</font></span></a></span>;<br/>
<a NAME="E35:71"/><span class="kw">then </span>
 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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="graph_5.html#E67"><span class="lab"><font color="Green" title="E52">Lm3</font></span></a></span>;<br/>
<a NAME="E36:71"/><span class="kw">then </span>
 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="c12">h</font> holds <br/><font color="Maroon" title="c12">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:71"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="txt" href="graph_5.html#E68"><span class="lab"><font color="Green" title="E53">Lm4</font></span></a></span>;<br/>
<a NAME="E37:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A18</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c12">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T84" title="RVSUM_1:th.84">RVSUM_1:84</a></span>;<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="c12">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="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c13">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:71"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="ref" href="rvsum_1.html#T75" title="RVSUM_1:th.75">RVSUM_1:75</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c12">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#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:71"><span class="lab"><font color="Green" title="E68">A15</font></span></a>, <a class="ref" href="finsop_1.html#T11" title="FINSOP_1:th.11">FINSOP_1:11</a></span>
;<br/>
<a NAME="E39:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A19</font></span>: 
 <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#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:71"><span class="lab"><font color="Green" title="E71">A18</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E40:71"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:71"><span class="lab"><font color="Green" title="E64">A11</font></span></a>, <a class="ref" href="rvsum_1.html#T75" title="RVSUM_1:th.75">RVSUM_1:75</a></span>;<br/>
<span class="kw">hence </span><a NAME="E41:71"/>
 <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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:71"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="txt" href="#E33:71"><span class="lab"><font color="Green" title="E70">A17</font></span></a>, <a class="txt" href="#E39:71"><span class="lab"><font color="Green" title="E72">A19</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</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>
