<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="graph_1.html#V7">finite</a> <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="graph_3.html#NM2">Vertex</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:64"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E2:64"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E3:64"/><i><font color="Green">E54</font></i>: 
the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="graph_3.html#E2:64"><i><font color="Green">E53</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">E55</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:64_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:64"><i><font color="Green">E52</font></i></a>, <a class="ref" href="graph_3.html#T44" target="_self">Th44</a></i>;<br/><a NAME="E3:64_1_1"/><i><font color="Green">E57</font></i>: 
(  <a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <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">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R1">misses</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">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:64"><i><font color="Green">E52</font></i></a>, <a class="txt" href="graph_3.html#E3:64"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_3.html#E1:64_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="graph_3.html#T46" target="_self">Th46</a></i>;<br/><b>thus </b> <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</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">E56</font></i></a>, <a class="txt" href="graph_3.html#E3:64_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1

;<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">E55</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:64_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:64"><i><font color="Green">E52</font></i></a>, <a class="ref" href="graph_3.html#T45" target="_self">Th45</a></i>;<br/><a NAME="E3:64_1_2"/><i><font color="Green">E57</font></i>: 
(  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <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">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R1">misses</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">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:64"><i><font color="Green">E52</font></i></a>, <a class="txt" href="graph_3.html#E3:64"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_3.html#E1:64_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="graph_3.html#T47" target="_self">Th47</a></i>;<br/><b>thus </b> <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graph_3.html#E2:64_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="graph_3.html#E3:64_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1

;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
