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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a></span>)</span><br/>  for <font color="Olive" title="b2">G9</font> being    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Olive" title="b1">S</font>  st <span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Olive" title="b1">S</font> is  <a href="orders_1.html#V3" title="ORDERS_1:attr.3">being_linear-order</a>  holds <br/> for <font color="Olive" title="b3">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Olive" title="b2">G9</font>  ex <font color="Olive" title="b4">H</font> being    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of <font color="Olive" title="b1">S</font> st <font color="Olive" title="b3">W</font> is    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Olive" title="b4">H</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G9</font> being    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c2">S</font>  st <span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">S</font> is  <a href="orders_1.html#V3" title="ORDERS_1:attr.3">being_linear-order</a>  holds <br/> for <font color="Olive" title="b2">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Olive" title="b1">G9</font>  ex <font color="Olive" title="b3">H</font> being    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Olive" title="b2">W</font> is    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Olive" title="b3">H</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">G9</font> be    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">S</font> is  <a href="orders_1.html#V3" title="ORDERS_1:attr.3">being_linear-order</a>  implies  for <font color="Olive" title="b1">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G9</font>  ex <font color="Olive" title="b2">H</font> being    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Olive" title="b1">W</font> is    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Olive" title="b2">H</font> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">R</font> = <span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">assume </span><a NAME="E1:63"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">S</font> is  <a href="orders_1.html#V3" title="ORDERS_1:attr.3">being_linear-order</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G9</font>  ex <font color="Olive" title="b2">H</font> being    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Olive" title="b1">W</font> is    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Olive" title="b2">H</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G9</font>] <span class="kw">means </span> ex <font color="Olive" title="b1">H</font> being    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> st $1 is    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Olive" title="b1">H</font>;<br/>
<a NAME="E2:63"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
 for <font color="Olive" title="b1">W</font> being  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>()  <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G9</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">W</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">W</font> be  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>()  <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G9</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">W</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">v9</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G9</font><span class="kw"> such that </span><br/><a NAME="E2:63_1"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
<font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G9</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c6">v9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T128" onmouseover="rs('glib_001/T128')" onmouseout="rh()">GLIB_001:128</a></span>;<br/>
<a NAME="E3:63_1"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="glib_014.html#K1" title="GLIB_014:func.1">the_Vertices_of</a> <font color="Maroon" title="c2">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_014.html#D25" onmouseover="rs('glib_014/D25')" onmouseout="rh()">GLIB_014:def 25</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">V</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E5:63_1"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
( <font color="Maroon" title="c6">v9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">V</font> &amp; <font color="Maroon" title="c7">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_014.html#K1" title="GLIB_014:func.1">the_Vertices_of</a> <font color="Maroon" title="c2">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">H</font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a><span class="kw"> such that </span><br/><a NAME="E7:63_1"/><span class="lab"><font color="Green" title="E44">A5</font></span>: 
( <font color="Maroon" title="c8">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c7">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c8">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="ref" href="glib_014.html#D14" onmouseover="rs('glib_014/D14')" onmouseout="rh()">GLIB_014:def 14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">H</font> = <font color="Maroon" title="c8">H</font> as    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:63_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">H</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">W</font> is    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c9">H</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">v</font> = <font color="Maroon" title="c6">v9</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c9">H</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="txt" href="#E7:63_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a></span>;<br/>
<font color="Maroon" title="c5">W</font> = 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</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="#E2:63_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="glib_001.html#D4" onmouseover="rs('glib_001/D4')" onmouseout="rh()">GLIB_001:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c9">H</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c10">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D4" onmouseover="rs('glib_001/D4')" onmouseout="rh()">GLIB_001:def 4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:63_1"/>
<font color="Maroon" title="c5">W</font> is    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c9">H</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:63"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
 for <font color="Olive" title="b1">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G9</font><br/>  for <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">W</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Olive" title="b2">e</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">W</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G9</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">W</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Olive" title="b1">e</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">W</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c6">e</font>] )</span><br/>



<span class="kw">assume </span><a NAME="E1:63_2"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
( <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">W</font>] )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c6">e</font>]</span><br/>

<a NAME="E2:63_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G9</font>
 
;<br/>
<a NAME="E3:63_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="glib_014.html#K2" title="GLIB_014:func.2">the_Edges_of</a> <font color="Maroon" title="c2">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_014.html#D25" onmouseover="rs('glib_014/D25')" onmouseout="rh()">GLIB_014:def 25</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E5:63_2"/><span class="lab"><font color="Green" title="E44">A8</font></span>: 
( <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E</font> &amp; <font color="Maroon" title="c7">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_014.html#K2" title="GLIB_014:func.2">the_Edges_of</a> <font color="Maroon" title="c2">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">H2</font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a><span class="kw"> such that </span><br/><a NAME="E7:63_2"/><span class="lab"><font color="Green" title="E45">A9</font></span>: 
( <font color="Maroon" title="c8">H2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c7">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c8">H2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_2"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="ref" href="glib_014.html#D15" onmouseover="rs('glib_014/D15')" onmouseout="rh()">GLIB_014:def 15</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">H2</font> = <font color="Maroon" title="c8">H2</font> as    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:63_2"><span class="lab"><font color="Green" title="E45">A9</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">H1</font> being    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E10:63_2"/><span class="lab"><font color="Green" title="E46">A10</font></span>: 
<font color="Maroon" title="c5">W</font> is    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c10">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_2"><span class="lab"><font color="Green" title="E43">A7</font></span></a></span>;<br/>
<a NAME="E11:63_2"/>
( <span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>  &amp; <span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="orders_1.html#D6" onmouseover="rs('orders_1/D6')" onmouseout="rh()">ORDERS_1:def 6</a></span>;<br/>
<a NAME="E12:63_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A11</font></span>: 
<span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">S</font> <a href="relat_2.html#R7" title="RELAT_2:pred.7">is_strongly_connected_in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D15" onmouseover="rs('relat_2/D15')" onmouseout="rh()">RELAT_2:def 15</a></span>;<br/>
<a NAME="E13:63_2"/><span class="lab"><font color="Green" title="E48">A12</font></span>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T40" target="_self" onmouseover="rs('glenum00/T40')" onmouseout="rh()">Th40</a></span>;<br/>
<a NAME="E14:63_2"/><span class="kw">then </span>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T71" onmouseover="rs('orders_1/T71')" onmouseout="rh()">ORDERS_1:71</a></span>;<br/>
<a NAME="E15:63_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A13</font></span>: 
( <font color="Maroon" title="c10">H1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">S</font></span>)</span> &amp; <font color="Maroon" title="c9">H2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">S</font></span>)</span> )
 
;<br/>
<a NAME="E16:63_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A14</font></span>: 
( <font color="Maroon" title="c10">H1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  &amp; <font color="Maroon" title="c9">H2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:63_2"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="ref" href="wellord1.html#T12" onmouseover="rs('wellord1/T12')" onmouseout="rh()">WELLORD1:12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G9</font><span class="kw"> such that </span><br/><a NAME="E18:63_2"/><span class="lab"><font color="Green" title="E51">A15</font></span>: 
<font color="Maroon" title="c6">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c3">G9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_2"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="ref" href="glib_000.html#T64" onmouseover="rs('glib_000/T64')" onmouseout="rh()">GLIB_000:64</a></span>;<br/>
<a NAME="E19:63_2"/><span class="lab"><font color="Green" title="E52">A16</font></span>: 
( <font color="Maroon" title="c10">H1</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c3">G9</font> &amp; <font color="Maroon" title="c9">H2</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c3">G9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_014.html#T21" onmouseover="rs('glib_014/T21')" onmouseout="rh()">GLIB_014:21</a></span>;<br/>
<a NAME="E20:63_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A17</font></span>: 
<font color="Maroon" title="c6">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c9">H2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_2"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="txt" href="#E7:63_2"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="txt" href="#E18:63_2"><span class="lab"><font color="Green" title="E51">A15</font></span></a>, <a class="ref" href="glib_000.html#T73" onmouseover="rs('glib_000/T73')" onmouseout="rh()">GLIB_000:73</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:63_2_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">H1</font>,<font color="Maroon" title="c9">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">S</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">H2</font>,<font color="Maroon" title="c10">H1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">S</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:63_2"><span class="lab"><font color="Green" title="E47">A11</font></span></a>, <a class="txt" href="#E15:63_2"><span class="lab"><font color="Green" title="E49">A13</font></span></a>, <a class="ref" href="relat_2.html#D7" onmouseover="rs('relat_2/D7')" onmouseout="rh()">RELAT_2:def 7</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:63_2_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">H1</font>,<font color="Maroon" title="c9">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c6">e</font>]</span><br/><div class="add"><a NAME="E2:63_2_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">H1</font>,<font color="Maroon" title="c9">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mmlquer2.html#T4" onmouseover="rs('mmlquer2/T4')" onmouseout="rh()">MMLQUER2:4</a></span>;<br/><a NAME="E3:63_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">H1</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c9">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:63_2"><span class="lab"><font color="Green" title="E50">A14</font></span></a>, <a class="ref" href="glenum00.html#D6" target="_self" onmouseover="rs('glenum00/D6')" onmouseout="rh()">Def6</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">W2</font> = <font color="Maroon" title="c5">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c9">H2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:63_2"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="ref" href="glib_001.html#T167" onmouseover="rs('glib_001/T167')" onmouseout="rh()">GLIB_001:167</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c9">H2</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c6">e</font> is    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c9">H2</font></span><br/><a NAME="E5:63_2_1_1"/>
<font color="Maroon" title="c5">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">W2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T16" onmouseover="rs('glib_001/T16')" onmouseout="rh()">GLIB_001:16</a></span>;<br/><a NAME="E6:63_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">W2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:63_2"><span class="lab"><font color="Green" title="E53">A17</font></span></a>, <a class="ref" href="glib_000.html#T62" onmouseover="rs('glib_000/T62')" onmouseout="rh()">GLIB_000:62</a></span>;<br/><a NAME="E7:63_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">W2</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c6">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:63_2"><span class="lab"><font color="Green" title="E52">A16</font></span></a>, <a class="ref" href="glib_001.html#T174" onmouseover="rs('glib_001/T174')" onmouseout="rh()">GLIB_001:174</a></span>;<br/><span class="kw">hence </span><a NAME="E8:63_2_1_1"/>
<font color="Maroon" title="c5">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c6">e</font> is    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c9">H2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="hsNdiv(this)" href="javascript:()" title="63_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:63_2_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">H2</font>,<font color="Maroon" title="c10">H1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c6">e</font>]</span><br/><div class="add"><a NAME="E2:63_2_1_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">H2</font>,<font color="Maroon" title="c10">H1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mmlquer2.html#T4" onmouseover="rs('mmlquer2/T4')" onmouseout="rh()">MMLQUER2:4</a></span>;<br/><a NAME="E3:63_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">H2</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c10">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:63_2"><span class="lab"><font color="Green" title="E50">A14</font></span></a>, <a class="ref" href="glenum00.html#D6" target="_self" onmouseover="rs('glenum00/D6')" onmouseout="rh()">Def6</a></span>;<br/><a NAME="E4:63_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A18</font></span>: 
<font color="Maroon" title="c6">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c10">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:63_2"><span class="lab"><font color="Green" title="E53">A17</font></span></a>, <a class="ref" href="glib_000.html#T72" onmouseover="rs('glib_000/T72')" onmouseout="rh()">GLIB_000:72</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">W1</font> = <font color="Maroon" title="c5">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c10">H1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:63_2"><span class="lab"><font color="Green" title="E46">A10</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c10">H1</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c6">e</font> is    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c10">H1</font></span><br/><a NAME="E6:63_2_1_2"/>
<font color="Maroon" title="c5">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">W1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T16" onmouseover="rs('glib_001/T16')" onmouseout="rh()">GLIB_001:16</a></span>;<br/><a NAME="E7:63_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">W1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_2_1_2"><span class="lab"><font color="Green" title="E54">A18</font></span></a>, <a class="ref" href="glib_000.html#T62" onmouseover="rs('glib_000/T62')" onmouseout="rh()">GLIB_000:62</a></span>;<br/><a NAME="E8:63_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c6">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:63_2"><span class="lab"><font color="Green" title="E52">A16</font></span></a>, <a class="ref" href="glib_001.html#T174" onmouseover="rs('glib_001/T174')" onmouseout="rh()">GLIB_001:174</a></span>;<br/><span class="kw">hence </span><a NAME="E9:63_2_1_2"/>
<font color="Maroon" title="c5">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c6">e</font> is    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c10">H1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E4:63"/>
 for <font color="Olive" title="b1">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G9</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">W</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="glib_009.html#S1" onmouseover="rs('glib_009/S1')" onmouseout="rh()">GLIB_009:sch 1</a>(<a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E3:63"><span class="lab"><font color="Green" title="E42">A6</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:63"/>
 for <font color="Olive" title="b1">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G9</font>  ex <font color="Olive" title="b2">H</font> being    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Olive" title="b1">W</font> is    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Olive" title="b2">H</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
