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

<b>let </b><font color="Maroon">G</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">v</font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">X2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X1</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:42"/>
<font color="Maroon">X2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font>
 ;<br/>

<a NAME="E2:42"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X2</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">X2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_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="E1:42_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X1</font>
 ;<br/><a NAME="E2:42_1_1"/><b>then </b><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> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i/>;<br/><a NAME="E3:42_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">X2</font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E2:42_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:42_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X2</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">X2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E5:42_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">X2</font></span>)</span></span>)</span>
 <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>assume </b><a NAME="E2:42_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">X2</font></span>)</span></span>)</span>
 ;<br/><a NAME="E3:42_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X2</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">X2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:42_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 <font color="Maroon">G</font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">X2</font> ) )
 <b>by </b><i/>;<br/><a NAME="E5:42_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">X2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:42_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X1</font>
 <b>by </b><i>, <a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E4:42"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X1</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">v</font>,<font color="Maroon">X2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">X2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E5:42"/>
 <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X2</font> <a href="xboole_0.html#R1">misses</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">X2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_2"/>
not  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X2</font> <a href="xboole_0.html#R1">misses</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">X2</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:42_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">X2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>
<a NAME="E4:42_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">X2</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:42_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">X2</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:42_2"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:42"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X1</font></span>)</span> <a href="hidden.html#R1">=</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#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X2</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#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<span class="p3">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">X2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>;<br/>
<b>hence </b><a NAME="E7:42"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</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#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X2</font></span>)</span></span>)</span>
 ;<br/>


</div>
