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

<span class="kw">let </span><font color="Maroon" title="c1">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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 <font color="Maroon" title="c1">G</font><br/>  for <font color="Olive" title="b2">X2</font>, <font color="Olive" title="b3">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">X1</font> holds <br/> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Olive" title="b1">v</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b2">X2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b3">X1</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">X2</font>)</span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">v</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X2</font>, <font color="Olive" title="b2">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">X1</font> holds <br/> <a href="card_1.html#K5" title="CARD_1:func.5">card</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="c2">v</font>,<span class="p2">(<span class="default"><font color="Olive" title="b2">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">X2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b2">X1</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">X2</font>)</span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">X2</font>, <font color="Maroon" title="c4">X1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X1</font> implies  <a href="card_1.html#K5" title="CARD_1:func.5">card</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="c2">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">X1</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X2</font>)</span>)</span></span>)</span> )</span><br/>







<span class="kw">assume </span><a NAME="E1:41"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c3">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</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="c2">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">X1</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X2</font>)</span>)</span></span>)</span></span><br/>

<a NAME="E2:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c4">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">X1</font>) implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">v</font>,<font color="Maroon" title="c3">X2</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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="c2">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font></span>)</span>)</span>)</span> ) &amp; ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">v</font>,<font color="Maroon" title="c3">X2</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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="c2">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font></span>)</span>)</span>)</span> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">X1</font>) ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">v</font>,<font color="Maroon" title="c3">X2</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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="c2">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font></span>)</span>)</span>)</span> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">X1</font>) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:41_1_1"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">X1</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">v</font>,<font color="Maroon" title="c3">X2</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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="c2">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font></span>)</span>)</span>)</span></span><br/><a NAME="E2:41_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#D1" target="_self" title="GRAPH_3:def.1">Def1</a></span>;<br/><a NAME="E3:41_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A4</font></span>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E4:41_1_1"/>
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="graph_3.html#D1" target="_self" title="GRAPH_3:def.1">Def1</a></span>;<br/><a NAME="E5:41_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X2</font>) or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E3:41_1_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="graph_3.html#D1" target="_self" title="GRAPH_3:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:41_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">v</font>,<font color="Maroon" title="c3">X2</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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="c2">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:41_1"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">v</font>,<font color="Maroon" title="c3">X2</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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="c2">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font></span>)</span>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">X1</font>)</span><br/><a NAME="E3:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A6</font></span>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X2</font>) or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E4:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A7</font></span>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#D1" target="_self" title="GRAPH_3:def.1">Def1</a></span>;<br/><a NAME="E5:41_1"/>
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="graph_3.html#D1" target="_self" title="GRAPH_3:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:41_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">X1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E4:41_1"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="graph_3.html#D1" target="_self" title="GRAPH_3:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A8</font></span>: 
 <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">X1</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="c2">v</font>,<font color="Maroon" title="c3">X2</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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="c2">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/>
<a NAME="E5:41"/>
 <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X2</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font></span>)</span>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:41_2"/>
 not  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X2</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font></span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:41_2"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">v</font>,<font color="Maroon" title="c3">X2</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="c2">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></span>;<br/>
<a NAME="E4:41_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:41_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A10</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#D1" target="_self" title="GRAPH_3:def.1">Def1</a></span>;<br/>
<a NAME="E6:41_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E7:41_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#D1" target="_self" title="GRAPH_3:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:41_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:41"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</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="c2">v</font>,<font color="Maroon" title="c4">X1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X2</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#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="card_2.html#T40" title="CARD_2:th.40">CARD_2:40</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:41"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</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="c2">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">X2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">X1</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X2</font>)</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
