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<b>let </b><font color="Maroon">G1</font> be   <a href="glib_000.html#NM2">_Graph</a>, <font color="Maroon">G2</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/>

<b>assume </b><a NAME="E1:58"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">G1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G2</font>
 ;<br/>

<b>let </b><font color="Maroon">S</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><i><font color="Green">E33</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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:58_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <font color="Maroon">S</font>
 ;<br/><b>reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">x</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G1</font> <b>by </b><i/>;<br/><a NAME="E3:58_1"/><i><font color="Green">E56</font></i>: 
( not <font color="Maroon">t1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">v</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G1</font> st <br/>( <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">t1</font>,<font color="Olive">v</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  ) )
 <b>by </b><i>, <a class="ref" href="chord.html#T11" target="_self">Th11</a></i>;<br/><b>consider </b><font color="Maroon">v1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E5:58_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">t1</font>,<font color="Maroon">v1</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  )
 <b>by </b><i>, <a class="ref" href="chord.html#T11" target="_self">Th11</a></i>;<br/><b>reconsider </b><font color="Maroon">t2</font> = <font color="Maroon">t1</font>, <font color="Maroon">v2</font> = <font color="Maroon">v1</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font> <b>by </b><i><a class="ref" href="chord.html#T13" target="_self">Th13</a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E7:58_1"/>
<font color="Maroon">t2</font>,<font color="Maroon">v2</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i><a class="ref" href="chord.html#T13" target="_self">Th13</a>, , <a class="ref" href="chord.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E8:58_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G2</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <font color="Maroon">S</font>
 <b>by </b><i>, </i>;<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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:58_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G2</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <font color="Maroon">S</font>
 ;<br/><b>reconsider </b><font color="Maroon">t2</font> = <font color="Maroon">x</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font> <b>by </b><i/>;<br/><a NAME="E3:58_2"/><i><font color="Green">E67</font></i>: 
( not <font color="Maroon">t2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">v</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font> st <br/>( <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">t2</font>,<font color="Olive">v</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  ) )
 <b>by </b><i>, <a class="ref" href="chord.html#T11" target="_self">Th11</a></i>;<br/><b>consider </b><font color="Maroon">v2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font><b> such that </b><br/><a NAME="E5:58_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">t2</font>,<font color="Maroon">v2</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  )
 <b>by </b><i>, <a class="ref" href="chord.html#T11" target="_self">Th11</a></i>;<br/><b>reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">t2</font>, <font color="Maroon">v1</font> = <font color="Maroon">v2</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G1</font> <b>by </b><i><a class="ref" href="chord.html#T13" target="_self">Th13</a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E7:58_2"/>
<font color="Maroon">t1</font>,<font color="Maroon">v1</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i><a class="ref" href="chord.html#T13" target="_self">Th13</a>, , <a class="ref" href="chord.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E8:58_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <font color="Maroon">S</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:58"/>
<font color="Maroon">G1</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E15"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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