<?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" 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">S</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon" title="c1">G1</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <font color="Olive" title="b1">S</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:57"/><span class="lab"><font color="Green" title="E48">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" 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    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon" title="c1">G1</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <font color="Olive" title="b1">S</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">S</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" 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="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <font color="Maroon" title="c3">S</font></span><br/>

<div><span class="lab"><font color="Green" title="E49">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <font color="Maroon" title="c3">S</font> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <font color="Maroon" title="c3">S</font> )</span><br/><span class="kw">assume </span><a NAME="E1:57_1"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <font color="Maroon" title="c3">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <font color="Maroon" title="c3">S</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c5">t2</font> = <font color="Maroon" title="c4">x</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1"><span class="lab"><font color="Green" title="E50">A3</font></span></a></span>;<br/><a NAME="E3:57_1"/><span class="lab"><font color="Green" title="E51">A4</font></span>: 
 not <font color="Maroon" title="c5">t2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="chord.html#T50" target="_self" title="CHORD:th.50">Th50</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font><span class="kw"> such that </span><br/><a NAME="E5:57_1"/><span class="lab"><font color="Green" title="E52">A5</font></span>: 
<font color="Maroon" title="c6">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">and </span><br/><a NAME="E6:57_1"/><span class="lab"><font color="Green" title="E53">A6</font></span>: 
<font color="Maroon" title="c5">t2</font>,<font color="Maroon" title="c6">v2</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="#E1:57_1"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="chord.html#T50" target="_self" title="CHORD:th.50">Th50</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">t1</font> = <font color="Maroon" title="c5">t2</font>, <font color="Maroon" title="c8">v1</font> = <font color="Maroon" title="c6">v2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="glib_000.html#D34" title="GLIB_000:def.34">GLIB_000:def 34</a></span>;<br/><a NAME="E8:57_1"/>
<font color="Maroon" title="c7">t1</font>,<font color="Maroon" title="c8">v1</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="#E1:57"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E6:57_1"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="ref" href="chord.html#T44" target="_self" title="CHORD:th.44">Th44</a></span>;<br/><span class="kw">hence </span><a NAME="E9:57_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E5:57_1"><span class="lab"><font color="Green" title="E52">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <font color="Maroon" title="c3">S</font> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <font color="Maroon" title="c3">S</font> )</span><br/><span class="kw">assume </span><a NAME="E1:57_2"/><span class="lab"><font color="Green" title="E50">A7</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <font color="Maroon" title="c3">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <font color="Maroon" title="c3">S</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c5">t1</font> = <font color="Maroon" title="c4">x</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_2"><span class="lab"><font color="Green" title="E50">A7</font></span></a></span>;<br/><a NAME="E3:57_2"/><span class="lab"><font color="Green" title="E51">A8</font></span>: 
 not <font color="Maroon" title="c5">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_2"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="ref" href="chord.html#T50" target="_self" title="CHORD:th.50">Th50</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">v1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E5:57_2"/><span class="lab"><font color="Green" title="E52">A9</font></span>: 
<font color="Maroon" title="c6">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">and </span><br/><a NAME="E6:57_2"/><span class="lab"><font color="Green" title="E53">A10</font></span>: 
<font color="Maroon" title="c5">t1</font>,<font color="Maroon" title="c6">v1</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="#E1:57_2"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="ref" href="chord.html#T50" target="_self" title="CHORD:th.50">Th50</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">t2</font> = <font color="Maroon" title="c5">t1</font>, <font color="Maroon" title="c8">v2</font> = <font color="Maroon" title="c6">v1</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="glib_000.html#D34" title="GLIB_000:def.34">GLIB_000:def 34</a></span>;<br/><a NAME="E8:57_2"/>
<font color="Maroon" title="c7">t2</font>,<font color="Maroon" title="c8">v2</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="#E1:57"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E6:57_2"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="ref" href="chord.html#T44" target="_self" title="CHORD:th.44">Th44</a></span>;<br/><span class="kw">hence </span><a NAME="E9:57_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_2"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="txt" href="#E5:57_2"><span class="lab"><font color="Green" title="E52">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:57"/>
<font color="Maroon" title="c1">G1</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
