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

<span class="kw">let </span><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</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"> ( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> implies  for <font color="Olive" title="b1">u1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b2">u2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u2</font> &amp; <font color="Olive" title="b1">u1</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  holds <br/><font color="Olive" title="b2">u2</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:75"/><span class="lab"><font color="Green" title="E62">A1</font></span>: 
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</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">u1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b2">u2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u2</font> &amp; <font color="Olive" title="b1">u1</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  holds <br/><font color="Olive" title="b2">u2</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">u1</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</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">u2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c3">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u2</font> &amp; <font color="Maroon" title="c3">u1</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  holds <br/><font color="Olive" title="b1">u2</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">u2</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</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="c3">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u2</font> &amp; <font color="Maroon" title="c3">u1</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  implies <font color="Maroon" title="c4">u2</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E2:75"/><span class="lab"><font color="Green" title="E63">A2</font></span>: 
<font color="Maroon" title="c3">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u2</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"> (  not <font color="Maroon" title="c3">u1</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  or <font color="Maroon" title="c4">u2</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:75"/><span class="lab"><font color="Green" title="E64">A3</font></span>: 
<font color="Maroon" title="c3">u1</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</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="c4">u2</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="75_1"><span class="kw">now </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="c4">u2</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_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="E1:75_1_1"/>
( <font color="Maroon" title="c1">G1</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">u1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c1">G1</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">u1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:75_1_1_1"/>
<font color="Maroon" title="c1">G1</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">u1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.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"> <font color="Maroon" title="c4">u2</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> </span><br/><div class="add"><a NAME="E2:75_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">G2</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="chord.html#T51" target="_self" onmouseover="rs('chord/T51')" onmouseout="rh()">Th50</a></span>;<br/><span class="kw">hence </span><a NAME="E3:75_1_1_1"/>
<font color="Maroon" title="c4">u2</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:75_1_1_2"/><span class="lab"><font color="Green" title="E65">A4</font></span>: 
<font color="Maroon" title="c1">G1</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">u1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.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"> <font color="Maroon" title="c4">u2</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">H1</font> =  the    <a href="chord.html#M1" title="CHORD:mode.1">AdjGraph</a> of <font color="Maroon" title="c1">G1</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">u1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><a NAME="E2:75_1_1_2"/>
 the    <a href="chord.html#M1" title="CHORD:mode.1">AdjGraph</a> of <font color="Maroon" title="c1">G1</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">u1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="txt" href="#E1:75_1_1_2"><span class="lab"><font color="Green" title="E65">A4</font></span></a></span>;<br/><a NAME="E3:75_1_1_2"/><span class="kw">then </span>
 for <font color="Olive" title="b1">H2</font> being    <a href="chord.html#M1" title="CHORD:mode.1">AdjGraph</a> of <font color="Maroon" title="c2">G2</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds  <font color="Olive" title="b1">H2</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="chord.html#T59" target="_self" onmouseover="rs('chord/T59')" onmouseout="rh()">Th58</a>, <a class="ref" href="chord.html#T62" target="_self" onmouseover="rs('chord/T62')" onmouseout="rh()">Th61</a></span>;<br/><span class="kw">hence </span><a NAME="E4:75_1_1_2"/>
<font color="Maroon" title="c4">u2</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:75"/>
<font color="Maroon" title="c4">u2</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</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"> verum</span><br/>


</div>
