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<b>let </b><font color="Maroon">G</font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">v2</font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>;<br/>





<b>set </b><font color="Maroon">G'</font> =  <a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>;<br/>
<b>set </b><font color="Maroon">E</font> = the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">S</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">T</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">E'</font> = the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span>;<br/>
<a NAME="E1:51"/><i><font color="Green">E30</font></i>: 
the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma38</font></i></a></i>;<br/>
<a NAME="E2:51"/>
the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:51"/>
the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_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/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:51_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 ;<br/><a NAME="E2:51_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:51_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><a NAME="E4:51_1_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:51_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="graph_3.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:51_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E3:51_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:51_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:51"/>
the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<a NAME="E6:51"/><i><font color="Green">E37</font></i>: 
( the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">v1</font></span>)</span> &amp; the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">v2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma38</font></i></a></i>;<br/>
<a NAME="E7:51"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">v1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E8:51"/><b>then </b>
the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">v1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b>the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span> <a href="funct_1.html#K1">.</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> = 
<span class="p1">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">v1</font></span>)</span> <a href="funct_1.html#K1">.</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">v1</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><br/>

<a NAME="E10:51"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">v2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E11:51"/><b>then </b>
the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">v2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b>the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span> <a href="funct_1.html#K1">.</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> = 
<span class="p1">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">v2</font></span>)</span> <a href="funct_1.html#K1">.</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">v2</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><br/>


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