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<b>let </b><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E1:58"/><i><font color="Green" title="E48">A1</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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><i><font color="Green" title="E49">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>assume </b><a NAME="E1:58_1"/><i><font color="Green" title="E50">A3</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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/><b>reconsider </b><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> <b>by </b><i><a class="txt" href="#E1:58_1"><i><font color="Green" title="E50">A3</font></i></a></i>;<br/><a NAME="E3:58_1"/><i><font color="Green" title="E51">A4</font></i>: 
( 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> &amp;  ex <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c5">t1</font>,<font color="Olive" title="b1">v</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) )
 <b>by </b><i><a class="txt" href="#E1:58_1"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="chord.html#T50" target="_self" title="CHORD:th.50">Th50</a></i>;<br/><b>consider </b><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><b> such that </b><br/><a NAME="E5:58_1"/><i><font color="Green" title="E52">A5</font></i>: 
( <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> &amp; <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>  )
 <b>by </b><i><a class="txt" href="#E1:58_1"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="chord.html#T50" target="_self" title="CHORD:th.50">Th50</a></i>;<br/><b>reconsider </b><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> <b>by </b><i><a class="txt" href="#E1:58"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/><a NAME="E7:58_1"/>
<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> 
 <b>by </b><i><a class="txt" href="#E1:58"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="#E5:58_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="chord.html#T44" target="_self" title="CHORD:th.44">Th44</a></i>;<br/><b>hence </b><a NAME="E8:58_1"/>
<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>
 <b>by </b><i><a class="txt" href="#E3:58_1"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="#E5:58_1"><i><font color="Green" title="E52">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>assume </b><a NAME="E1:58_2"/><i><font color="Green" title="E50">A6</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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/><b>reconsider </b><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> <b>by </b><i><a class="txt" href="#E1:58_2"><i><font color="Green" title="E50">A6</font></i></a></i>;<br/><a NAME="E3:58_2"/><i><font color="Green" title="E51">A7</font></i>: 
( 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> &amp;  ex <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c5">t2</font>,<font color="Olive" title="b1">v</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) )
 <b>by </b><i><a class="txt" href="#E1:58_2"><i><font color="Green" title="E50">A6</font></i></a>, <a class="ref" href="chord.html#T50" target="_self" title="CHORD:th.50">Th50</a></i>;<br/><b>consider </b><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><b> such that </b><br/><a NAME="E5:58_2"/><i><font color="Green" title="E52">A8</font></i>: 
( <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> &amp; <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>  )
 <b>by </b><i><a class="txt" href="#E1:58_2"><i><font color="Green" title="E50">A6</font></i></a>, <a class="ref" href="chord.html#T50" target="_self" title="CHORD:th.50">Th50</a></i>;<br/><b>reconsider </b><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> <b>by </b><i><a class="txt" href="#E1:58"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/><a NAME="E7:58_2"/>
<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> 
 <b>by </b><i><a class="txt" href="#E1:58"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="#E5:58_2"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="chord.html#T44" target="_self" title="CHORD:th.44">Th44</a></i>;<br/><b>hence </b><a NAME="E8:58_2"/>
<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>
 <b>by </b><i><a class="txt" href="#E3:58_2"><i><font color="Green" title="E51">A7</font></i></a>, <a class="txt" href="#E5:58_2"><i><font color="Green" title="E52">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:58"/>
<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>
 <b>by </b><i><a class="txt" href="#E2:58"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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