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<b>let </b><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font> be   <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W1</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b2">W2</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W2</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c2">G2</font> holds <br/><font color="Olive" title="b1">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W2</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span><br/><b>let </b><font color="Maroon" title="c3">W1</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W2</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W2</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c2">G2</font> holds <br/><font color="Maroon" title="c3">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W2</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span><br/><b>let </b><font color="Maroon" title="c4">W2</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c2">G2</font> implies <font color="Maroon" title="c3">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a>  )</span><br/>





<b>assume </b><a NAME="E1:167"/><i><font color="Green" title="E35">A1</font></i>: 
( <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c2">G2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span><br/>

<a NAME="E2:167"/><b>then </b><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#T87" title="GLIB_001:th.87">GLIB_001:87</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">WS1</font> = <font color="Maroon" title="c3">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> ;<br/>
<b>set </b><font color="Maroon" title="c6">WS2</font> = <font color="Maroon" title="c4">W2</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="167_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:167_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </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">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </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">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</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"><font color="Maroon" title="c3">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span><br/><b>thus </b><i><font color="Green" title="E37">A3</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span>
<b>by </b><i><a class="txt" href="glib_003.html#E2:167"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="glib_003.html#D18" target="_self" title="GLIB_003:def.18">Def18</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span>
<b>by </b><i><a class="ref" href="glib_003.html#D18" target="_self" title="GLIB_003:def.18">Def18</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</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"><font color="Maroon" title="c3">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><b>let </b><font color="Maroon" title="c7">x</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">x</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"><font color="Maroon" title="c3">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )</span><br/><b>assume </b><a NAME="E3:167_1"/>
<font color="Maroon" title="c7">x</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"><font color="Maroon" title="c3">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><a NAME="E4:167_1"/><b>then </b><i><font color="Green" title="E38">A4</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</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"><font color="Maroon" title="c3">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> &amp; <font color="Maroon" title="c7">x</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"><font color="Maroon" title="c4">W2</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="glib_003.html#E2:167_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
<span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c2">G2</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"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="ref" href="glib_003.html#D18" target="_self" title="GLIB_003:def.18">Def18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</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"><font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="txt" href="glib_003.html#E1:167"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="glib_001.html#T87" title="GLIB_001:th.87">GLIB_001:87</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<b>by </b><i><a class="txt" href="glib_003.html#E4:167_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="glib_003.html#D18" target="_self" title="GLIB_003:def.18">Def18</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:167"/>
<font color="Maroon" title="c3">W1</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> 
 <b>by </b><i><a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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