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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">G</font> be  <a href="graph_1.html#V7" title="GRAPH_1:attr.7">finite</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c5">v2</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c2">G</font>;<br/><b>let </b><font color="Maroon" title="c6">v'</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font></span>)</span>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:64"/><i><font color="Green" title="E52">A1</font></i>: 
( <font color="Maroon" title="c6">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c4">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v2</font> )
 <b>and </b><br/><a NAME="E2:64"/><i><font color="Green" title="E53">A2</font></i>: 
( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font> or <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> )
 <b>and </b><br/><a NAME="E3:64"/><i><font color="Green" title="E54">A3</font></i>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c7">E</font> = the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1"/>
( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font> or <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> )
 </span><b>by </b><i><a class="txt" href="graph_3.html#E2:64"><i><font color="Green" title="E53">A2</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1"><b>suppose </b></a><a NAME="E1:64_1_1"/><i><font color="Green" title="E55">A4</font></i>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font>
 ;<br/><div class="add"><a NAME="E2:64_1_1"/><b>then </b><i><font color="Green" title="E56">A5</font></i>: 
 <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> <font color="Maroon" title="c6">v'</font>,<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:64"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="graph_3.html#T44" target="_self" title="GRAPH_3:th.44">Th44</a></i>;<br/><a NAME="E3:64_1_1"/>
(  <a href="graph_3.html#K3" title="GRAPH_3:func.3">Edges_Out</a> <font color="Maroon" title="c6">v'</font>,<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_3.html#K3" title="GRAPH_3:func.3">Edges_Out</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="graph_3.html#K3" title="GRAPH_3:func.3">Edges_Out</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:64"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="graph_3.html#E3:64"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="graph_3.html#E1:64_1_1"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="graph_3.html#T46" target="_self" title="GRAPH_3:th.46">Th46</a></i>;<br/><b>hence </b> <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c6">v'</font>,<font color="Maroon" title="c1">X</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p3">(<span class="default"><a href="graph_3.html#K3" title="GRAPH_3:func.3">Edges_Out</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graph_3.html#E2:64_1_1"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p3">(<span class="default"><a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p3">(<span class="default"><a href="graph_3.html#K3" title="GRAPH_3:func.3">Edges_Out</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T79" title="CARD_1:th.79">CARD_1:79</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_2"><b>suppose </b></a><a NAME="E1:64_1_2"/><i><font color="Green" title="E55">A6</font></i>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font>
 ;<br/><div class="add"><a NAME="E2:64_1_2"/><b>then </b><i><font color="Green" title="E56">A7</font></i>: 
 <a href="graph_3.html#K3" title="GRAPH_3:func.3">Edges_Out</a> <font color="Maroon" title="c6">v'</font>,<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_3.html#K3" title="GRAPH_3:func.3">Edges_Out</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:64"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="graph_3.html#T45" target="_self" title="GRAPH_3:th.45">Th45</a></i>;<br/><a NAME="E3:64_1_2"/>
(  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> <font color="Maroon" title="c6">v'</font>,<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:64"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="graph_3.html#E3:64"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="graph_3.html#E1:64_1_2"><i><font color="Green" title="E55">A6</font></i></a>, <a class="ref" href="graph_3.html#T47" target="_self" title="GRAPH_3:th.47">Th47</a></i>;<br/><b>hence </b> <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c6">v'</font>,<font color="Maroon" title="c1">X</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p3">(<span class="default"><a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="graph_3.html#K3" title="GRAPH_3:func.3">Edges_Out</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graph_3.html#E2:64_1_2"><i><font color="Green" title="E56">A7</font></i></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p3">(<span class="default"><a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p3">(<span class="default"><a href="graph_3.html#K3" title="GRAPH_3:func.3">Edges_Out</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T79" title="CARD_1:th.79">CARD_1:79</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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