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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> ; <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">G</font> being <a href="graph_1.html#V6" title="GRAPH_1:attr.6">finite</a> <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a><br/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">v1</font>, <font color="Olive" title="b4">v2</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Olive" title="b1">G</font><br/> for <font color="Olive" title="b5">v'</font> being  <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="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font></span>)</span>  st <font color="Olive" title="b5">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">v1</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">v2</font> holds <br/> <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Olive" title="b5">v'</font>,<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Olive" title="b2">v</font>,<font color="Maroon" title="c1">X</font></span><br/><b>let </b><font color="Maroon" title="c2">G</font> be  <a href="graph_1.html#V6" title="GRAPH_1:attr.6">finite</a> <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a>; <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">v</font>, <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c2">G</font><br/> for <font color="Olive" title="b4">v'</font> being  <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="Olive" title="b2">v1</font>,<font color="Olive" title="b3">v2</font></span>)</span>  st <font color="Olive" title="b4">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v1</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">v2</font> holds <br/> <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Olive" title="b4">v'</font>,<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Olive" title="b1">v</font>,<font color="Maroon" title="c1">X</font></span><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>; <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">v'</font> being  <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>  st <font color="Olive" title="b1">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v1</font> &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v2</font> holds <br/> <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Olive" title="b1">v'</font>,<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v1</font> &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v2</font> implies  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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><br/>











<b>assume </b><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="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v1</font> &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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><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#K5" title="CARD_1:func.5">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#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">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="c6">v'</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="graph_3.html#T48" target="_self" title="GRAPH_3:th.48">Th48</a></i>
<br/>.= 
 <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>
<b>by </b><i><a class="txt" href="#E1:64"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="graph_3.html#T49" target="_self" title="GRAPH_3:th.49">Th49</a></i>
;<br/> <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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