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<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">G0</font> being    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font><br/>  for <font color="Olive" title="b2">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"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span><br/>  for <font color="Olive" title="b3">x</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><br/>  for <font color="Olive" title="b4">G1</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b2">S</font><br/>  for <font color="Olive" title="b5">G2</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>  st <font color="Olive" title="b4">G1</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  &amp; <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Olive" title="b4">G1</font></span>)</span> holds <br/><font color="Olive" title="b5">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">G0</font> be    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</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">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"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span><br/>  for <font color="Olive" title="b2">x</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><br/>  for <font color="Olive" title="b3">G1</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b4">G2</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>  st <font color="Olive" title="b3">G1</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  &amp; <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Olive" title="b3">G1</font></span>)</span> holds <br/><font color="Olive" title="b4">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">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"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></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">x</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><br/>  for <font color="Olive" title="b2">G1</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">S</font><br/>  for <font color="Olive" title="b3">G2</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>  st <font color="Olive" title="b2">G1</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  &amp; <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Olive" title="b2">G1</font></span>)</span> holds <br/><font color="Olive" title="b3">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</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">G1</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">S</font><br/>  for <font color="Olive" title="b2">G2</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>  st <font color="Olive" title="b1">G1</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Olive" title="b1">G1</font></span>)</span> holds <br/><font color="Olive" title="b2">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">G1</font> be   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">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">  for <font color="Olive" title="b1">G2</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>  st <font color="Maroon" title="c5">G1</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c5">G1</font></span>)</span> holds <br/><font color="Olive" title="b1">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">G2</font> be   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></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"> ( <font color="Maroon" title="c5">G1</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c5">G1</font></span>)</span> implies <font color="Maroon" title="c6">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:62"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
<font color="Maroon" title="c5">G1</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 <span class="kw">and </span><br/><a NAME="E2:62"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c5">G1</font></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"> <font color="Maroon" title="c6">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/>

<a NAME="E3:62"/><span class="lab"><font color="Green" title="E54">A3</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c5">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">xs</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E5:62"/><span class="lab"><font color="Green" title="E55">A4</font></span>: 
<font color="Maroon" title="c7">xs</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">and </span><br/><a NAME="E6:62"/><span class="lab"><font color="Green" title="E56">A5</font></span>: 
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">xs</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="chord.html#T50" target="_self" onmouseover="rs('chord/T50')" onmouseout="rh()">Th49</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E8:62"/><span class="lab"><font color="Green" title="E57">A6</font></span>: 
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">xs</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E56">A5</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">Sx</font> = <font color="Maroon" title="c3">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c10">u</font>, <font color="Maroon" title="c11">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c6">G2</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="glib_002.html#D1" onmouseover="rs('glib_002/D1')" onmouseout="rh()">GLIB_002:def 1</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">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c6">G2</font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c11">v</font></span><br/>

<a NAME="E10:62"/><span class="lab"><font color="Green" title="E58">A7</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c6">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Sx</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<a NAME="E11:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A8</font></span>: 
( <font color="Maroon" title="c10">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S</font> or <font color="Maroon" title="c10">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E12:62"/>
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E13:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A9</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">Sx</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E14:62"/><span class="lab"><font color="Green" title="E61">A10</font></span>: 
<font color="Maroon" title="c7">xs</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">Sx</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E15:62"/>
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c7">xs</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:62"><span class="lab"><font color="Green" title="E57">A6</font></span></a></span>;<br/>
<a NAME="E16:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A11</font></span>: 
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c7">xs</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:62"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E14:62"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="ref" href="chord.html#T19" target="_self" onmouseover="rs('chord/T19')" onmouseout="rh()">Th19</a></span>;<br/>
<a NAME="E17:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A12</font></span>: 
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">xs</font>,<font color="Maroon" title="c6">G2</font>
 
;<br/>
<a NAME="E18:62"/><span class="lab"><font color="Green" title="E64">A13</font></span>: 
( <font color="Maroon" title="c11">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S</font> or <font color="Maroon" title="c11">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:62"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E19:62"/><span class="lab"><font color="Green" title="E65">A14</font></span>: 
<font color="Maroon" title="c5">G1</font> is   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c6">G2</font>,<font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T30" target="_self" onmouseover="rs('chord/T30')" onmouseout="rh()">Th30</a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_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="E1:62_1"/>
( ( <font color="Maroon" title="c10">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c11">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S</font> ) or ( <font color="Maroon" title="c10">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> ) or ( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c11">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S</font> ) or ( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:62"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E18:62"><span class="lab"><font color="Green" title="E64">A13</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1"><span class="kw">suppose </span></a><a NAME="E1:62_1_1"/><span class="lab"><font color="Green" title="E66">A15</font></span>: 
( <font color="Maroon" title="c10">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c11">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">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">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c6">G2</font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c11">v</font></span><br/><div class="add"><a NAME="E2:62_1_1"/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c5">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c5">G1</font><span class="kw"> such that </span><br/><a NAME="E4:62_1_1"/><span class="lab"><font color="Green" title="E67">A16</font></span>: 
<font color="Maroon" title="c12">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c11">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E1:62_1_1"><span class="lab"><font color="Green" title="E66">A15</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">W</font> = <font color="Maroon" title="c12">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c6">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:62"><span class="lab"><font color="Green" title="E65">A14</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="c13">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" title="c13">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c11">v</font></span><br/><span class="kw">thus </span><a NAME="E6:62_1_1"/>
<font color="Maroon" title="c13">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c11">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_1_1"><span class="lab"><font color="Green" title="E67">A16</font></span></a></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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_2"><span class="kw">suppose </span></a><a NAME="E1:62_1_2"/><span class="lab"><font color="Green" title="E66">A17</font></span>: 
( <font color="Maroon" title="c10">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</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">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c6">G2</font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c11">v</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c5">G1</font><span class="kw"> such that </span><br/><a NAME="E3:62_1_2"/><span class="lab"><font color="Green" title="E67">A18</font></span>: 
<font color="Maroon" title="c12">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c7">xs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E5:62"><span class="lab"><font color="Green" title="E55">A4</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">W</font> = <font color="Maroon" title="c12">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c6">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:62"><span class="lab"><font color="Green" title="E65">A14</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="c13">W</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c7">xs</font>,<font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c4">x</font>)</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"> <font color="Maroon" title="c13">W</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c7">xs</font>,<font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c11">v</font></span><br/><a NAME="E5:62_1_2"/><span class="lab"><font color="Green" title="E68">A19</font></span>: 
<font color="Maroon" title="c6">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c7">xs</font>,<font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c4">x</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c7">xs</font>,<font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:62"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="ref" href="glib_001.html#T15" onmouseover="rs('glib_001/T15')" onmouseout="rh()">GLIB_001:15</a></span>;<br/><a NAME="E6:62_1_2"/>
<font color="Maroon" title="c13">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c7">xs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_2"><span class="lab"><font color="Green" title="E67">A18</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:62_1_2"/>
<font color="Maroon" title="c13">W</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c7">xs</font>,<font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c11">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_2"><span class="lab"><font color="Green" title="E66">A17</font></span></a>, <a class="txt" href="#E5:62_1_2"><span class="lab"><font color="Green" title="E68">A19</font></span></a>, <a class="ref" href="glib_001.html#T31" onmouseover="rs('glib_001/T31')" onmouseout="rh()">GLIB_001:31</a></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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_3"><span class="kw">suppose </span></a><a NAME="E1:62_1_3"/><span class="lab"><font color="Green" title="E66">A20</font></span>: 
( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c11">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">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">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c6">G2</font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c11">v</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c5">G1</font><span class="kw"> such that </span><br/><a NAME="E3:62_1_3"/><span class="lab"><font color="Green" title="E67">A21</font></span>: 
<font color="Maroon" title="c12">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c7">xs</font>,<font color="Maroon" title="c11">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E5:62"><span class="lab"><font color="Green" title="E55">A4</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">W</font> = <font color="Maroon" title="c12">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c6">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:62"><span class="lab"><font color="Green" title="E65">A14</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>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c7">xs</font>)</span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c13">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c7">xs</font>)</span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c13">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c11">v</font></span><br/><a NAME="E5:62_1_3"/><span class="lab"><font color="Green" title="E68">A22</font></span>: 
<font color="Maroon" title="c6">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c7">xs</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">xs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:62"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="ref" href="glib_001.html#T15" onmouseover="rs('glib_001/T15')" onmouseout="rh()">GLIB_001:15</a></span>;<br/><a NAME="E6:62_1_3"/>
<font color="Maroon" title="c13">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c7">xs</font>,<font color="Maroon" title="c11">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_3"><span class="lab"><font color="Green" title="E67">A21</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:62_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c7">xs</font>)</span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c13">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c11">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_3"><span class="lab"><font color="Green" title="E66">A20</font></span></a>, <a class="txt" href="#E5:62_1_3"><span class="lab"><font color="Green" title="E68">A22</font></span></a>, <a class="ref" href="glib_001.html#T31" onmouseover="rs('glib_001/T31')" onmouseout="rh()">GLIB_001:31</a></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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_4"><span class="kw">suppose </span></a><a NAME="E1:62_1_4"/>
( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</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">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c6">G2</font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c11">v</font></span><br/><div class="add"><a NAME="E2:62_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c6">G2</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c10">u</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c11">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T13" onmouseover="rs('glib_001/T13')" onmouseout="rh()">GLIB_001:13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:62_1_4"/>
 ex <font color="Olive">b<sub>1</sub></font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c6">G2</font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c11">v</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>

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