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

<b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E19">A1</font></i>: 
( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">X</font> )
 ;<br/>

<a NAME="E2:20"/><b>then </b><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:20"/><b>then </b>
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/>
<a NAME="E4:20"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
( 2 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E1:20"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:20"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="combgras.html#E2:20"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/>
<a NAME="E6:20"/><i><font color="Green" title="E23">A5</font></i>: 
the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="combgras.html#E4:20"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></i>;<br/>
<a NAME="E7:20"/><i><font color="Green" title="E24">A6</font></i>: 
the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 </span>}</span> 
 
<b>by </b><i><a class="txt" href="combgras.html#E4:20"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></i>;<br/>
<b>let </b><font color="Maroon" title="c3">K</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>;<br/>

<a NAME="E8:20"/><i><font color="Green" title="E25">A7</font></i>: 
(  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2 &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="combgras.html#E2:20"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="combgras.html#E5:20"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a>, <a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></i>;<br/>
<a NAME="E9:20"/><i><font color="Green" title="E26">A8</font></i>: 
( <font color="Maroon" title="c3">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> implies <font color="Maroon" title="c3">K</font> is <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1"/>
<font color="Maroon" title="c3">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">S</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E3:20_1"/><i><font color="Green" title="E27">A9</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> : (  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="combgras.html#D4" target="_self" title="COMBGRAS:def.4">Def4</a></i>;<br/>
<a NAME="E4:20_1"/><i><font color="Green" title="E28">A10</font></i>: 
<font color="Maroon" title="c4">S</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="combgras.html#E5:20"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="combgras.html#E8:20"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="combgras.html#E3:20_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a></i>;<br/>
<a NAME="E5:20_1"/><i><font color="Green" title="E29">A11</font></i>: 
<font color="Maroon" title="c3">K</font> is <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">COMBGRAS:def 2</a><br/>

<b>assume </b><a NAME="E1:20_1_1"/><i><font color="Green" title="E30">A12</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">A1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E3:20_1_1"/><i><font color="Green" title="E31">A13</font></i>: 
( <font color="Maroon" title="c7">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c7">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A1</font> )
 <b>by </b><i><a class="txt" href="combgras.html#E3:20_1"><i><font color="Green" title="E27">A9</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">B1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E5:20_1_1"/><i><font color="Green" title="E32">A14</font></i>: 
( <font color="Maroon" title="c8">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c8">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B1</font> )
 <b>by </b><i><a class="txt" href="combgras.html#E3:20_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="combgras.html#E1:20_1_1"><i><font color="Green" title="E30">A12</font></i></a></i>;<br/>
<a NAME="E6:20_1_1"/><i><font color="Green" title="E33">A15</font></i>: 
( <font color="Maroon" title="c5">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c6">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E8:20"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="combgras.html#E3:20_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="combgras.html#E5:20_1_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E7:20_1_1"/><b>then </b><i><font color="Green" title="E34">A16</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="combgras.html#E5:20_1_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<a NAME="E8:20_1_1"/>
<font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="combgras.html#E5:20_1_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E9:20_1_1"/><b>then </b><i><font color="Green" title="E35">A17</font></i>: 
( <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="combgras.html#E3:20_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="combgras.html#E6:20_1_1"><i><font color="Green" title="E33">A15</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E10:20_1_1"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="combgras.html#E5:20"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></i>;<br/>
<a NAME="E11:20_1_1"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 or <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T13" title="ORDINAL1:th.13">ORDINAL1:13</a></i>;<br/>
<a NAME="E12:20_1_1"/><b>then </b><i><font color="Green" title="E36">A18</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 or <font color="Maroon" title="c1">k</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="combgras.html#E8:20"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></i>;<br/>
<a NAME="E13:20_1_1"/><i><font color="Green" title="E37">A19</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 implies  ex <font color="Olive" title="b1">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_1_1"/><i><font color="Green" title="E38">A20</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 ;<br/>

<a NAME="E2:20_1_1_1"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="combgras.html#E5:20_1_1"><i><font color="Green" title="E32">A14</font></i></a></i>;<br/>
<a NAME="E3:20_1_1_1"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E5:20"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="combgras.html#E3:20_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="combgras.html#E5:20_1_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="combgras.html#E1:20_1_1_1"><i><font color="Green" title="E38">A20</font></i></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E4:20_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="combgras.html#E7:20"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">L</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E6:20_1_1_1"/><i><font color="Green" title="E39">A21</font></i>: 
<font color="Maroon" title="c9">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">B</font>
 ;<br/>
<a NAME="E7:20_1_1_1"/>
( <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E8:20_1_1_1"/><b>then </b>
( <font color="Maroon" title="c5">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L</font> &amp; <font color="Maroon" title="c6">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E4:20"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="combgras.html#E6:20_1_1_1"><i><font color="Green" title="E39">A21</font></i></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></i>;<br/>
<a NAME="E9:20_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c9">L</font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></i>;<br/>
<b>hence </b><a NAME="E10:20_1_1_1"/>
 ex <font color="Olive" title="b1">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E14:20_1_1"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> implies  ex <font color="Olive" title="b1">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 ;<br/>

<a NAME="E2:20_1_1_2"/><b>then </b><i><font color="Green" title="E38">A22</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="combgras.html#E5:20_1_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="combgras.html#E6:20_1_1"><i><font color="Green" title="E33">A15</font></i></a>, <a class="txt" href="combgras.html#E7:20_1_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a></i>;<br/>
<a NAME="E3:20_1_1_2"/><i><font color="Green" title="E39">A23</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">A</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E4:20_1_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_1_2_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<a NAME="E2:20_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E4:20"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="combgras.html#E8:20"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="combgras.html#E3:20_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></i>;<br/>
<b>hence </b><a NAME="E3:20_1_1_2_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:20_1_1_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">A</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_1_1_2"><i><font color="Green" title="E39">A23</font></i></a>, <a class="ref" href="card_1.html#T13" title="CARD_1:th.13">CARD_1:13</a></i>;<br/>
<a NAME="E6:20_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T4" title="CARD_2:th.4">CARD_2:4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:20_1_1_2"/><i><font color="Green" title="E40">A24</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E9:20_1_1_2"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E8:20_1_1_2"><i><font color="Green" title="E40">A24</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E10:20_1_1_2"/><b>then </b><i><font color="Green" title="E41">A25</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="combgras.html#E6:20_1_1"><i><font color="Green" title="E33">A15</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E11:20_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">A</font> <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"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E12:20_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">A</font> <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"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="combgras.html#E7:20"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="combgras.html#E10:20_1_1_2"><i><font color="Green" title="E41">A25</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">L</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E14:20_1_1_2"/><i><font color="Green" title="E42">A26</font></i>: 
<font color="Maroon" title="c10">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> <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"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>
<a NAME="E15:20_1_1_2"/>
( <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <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"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <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"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="combgras.html#E2:20_1_1_2"><i><font color="Green" title="E38">A22</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E16:20_1_1_2"/><b>then </b>
( <font color="Maroon" title="c5">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">L</font> &amp; <font color="Maroon" title="c6">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">L</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E4:20"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="combgras.html#E14:20_1_1_2"><i><font color="Green" title="E42">A26</font></i></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></i>;<br/>
<a NAME="E17:20_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c10">L</font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></i>;<br/>
<b>hence </b><a NAME="E18:20_1_1_2"/>
 ex <font color="Olive" title="b1">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:20_1_1"/>
 ex <font color="Olive" title="b1">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font>
 <b>by </b><i><a class="txt" href="combgras.html#E9:20_1_1"><i><font color="Green" title="E35">A17</font></i></a>, <a class="txt" href="combgras.html#E12:20_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="combgras.html#E13:20_1_1"><i><font color="Green" title="E37">A19</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:20_1"/>
for <font color="Olive" title="b1">U</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">U</font> is <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> &amp; <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U</font> holds <br/><font color="Olive" title="b1">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">U</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:20_1_2"/><i><font color="Green" title="E30">A27</font></i>: 
( <font color="Maroon" title="c5">U</font> is <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> &amp; <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">U</font> &amp; <font color="Maroon" title="c5">U</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">K</font> )
 ;<br/>

<a NAME="E2:20_1_2"/><b>then </b>
not <font color="Maroon" title="c5">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:20_1_2"/><i><font color="Green" title="E31">A28</font></i>: 
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">U</font> &amp; not <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:20_1_2"/><i><font color="Green" title="E32">A29</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="combgras.html#E4:20_1_2"><i><font color="Green" title="E31">A28</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">A2</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E7:20_1_2"/><i><font color="Green" title="E33">A30</font></i>: 
<font color="Maroon" title="c7">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font>
 <b>by </b><i><a class="txt" href="combgras.html#E4:20_1_2"><i><font color="Green" title="E31">A28</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">A1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E9:20_1_2"/><i><font color="Green" title="E34">A31</font></i>: 
( <font color="Maroon" title="c8">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c8">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 <b>by </b><i><a class="txt" href="combgras.html#E6:20"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="combgras.html#E5:20_1_2"><i><font color="Green" title="E32">A29</font></i></a></i>;<br/>
<a NAME="E10:20_1_2"/><i><font color="Green" title="E35">A32</font></i>: 
( not <font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c6">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E8:20"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="combgras.html#E3:20_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="combgras.html#E4:20_1_2"><i><font color="Green" title="E31">A28</font></i></a>, <a class="txt" href="combgras.html#E9:20_1_2"><i><font color="Green" title="E34">A31</font></i></a>, <a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a></i>;<br/>
<a NAME="E11:20_1_2"/><i><font color="Green" title="E36">A33</font></i>: 
( <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font> &amp; <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E12:20_1_2"/><i><font color="Green" title="E37">A34</font></i>: 
<font color="Maroon" title="c4">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="combgras.html#E10:20_1_2"><i><font color="Green" title="E35">A32</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E13:20_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E14:20_1_2"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )
 
<b>by </b><i><a class="txt" href="combgras.html#E2:20"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="combgras.html#E5:20"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></i>;<br/>
<a NAME="E15:20_1_2"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> or  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T13" title="ORDINAL1:th.13">ORDINAL1:13</a></i>;<br/>
<a NAME="E16:20_1_2"/><b>then </b><i><font color="Green" title="E38">A35</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2
 
<b>by </b><i><a class="txt" href="combgras.html#E2:20"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="combgras.html#E3:20_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="combgras.html#E4:20_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="combgras.html#E10:20_1_2"><i><font color="Green" title="E35">A32</font></i></a>, <a class="txt" href="combgras.html#E11:20_1_2"><i><font color="Green" title="E36">A33</font></i></a>, <a class="txt" href="combgras.html#E12:20_1_2"><i><font color="Green" title="E37">A34</font></i></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></i>;<br/>
<a NAME="E17:20_1_2"/>
<font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="txt" href="combgras.html#E5:20"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E18:20_1_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">S</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">A</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E5:20"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="combgras.html#E8:20"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="combgras.html#E3:20_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="combgras.html#E9:20_1_2"><i><font color="Green" title="E34">A31</font></i></a>, <a class="ref" href="card_2.html#T58" title="CARD_2:th.58">CARD_2:58</a></i>;<br/>
<a NAME="E19:20_1_2"/><b>then </b>
<font color="Maroon" title="c6">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T4" title="CARD_2:th.4">CARD_2:4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E21:20_1_2"/><i><font color="Green" title="E39">A36</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E22:20_1_2"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> &amp; not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E21:20_1_2"><i><font color="Green" title="E39">A36</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E23:20_1_2"/><b>then </b><i><font color="Green" title="E40">A37</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="combgras.html#E4:20_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="combgras.html#E9:20_1_2"><i><font color="Green" title="E34">A31</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E24:20_1_2"/><b>then </b><i><font color="Green" title="E41">A38</font></i>: 
<font color="Maroon" title="c4">S</font> <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"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E25:20_1_2"/><i><font color="Green" title="E42">A39</font></i>: 
not <font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_2_1"/>
<font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E2:20_1_2_1"/><b>then </b>
( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">S</font> &amp; <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E9:20_1_2"><i><font color="Green" title="E34">A31</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/>
<a NAME="E3:20_1_2_1"/><b>then </b><i><font color="Green" title="E43">A40</font></i>: 
( <font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font> &amp; <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>, <a class="ref" href="xboole_1.html#T43" title="XBOOLE_1:th.43">XBOOLE_1:43</a></i>;<br/>
<a NAME="E4:20_1_2_1"/>
<font color="Maroon" title="c4">S</font> <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"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="combgras.html#E6:20"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="combgras.html#E23:20_1_2"><i><font color="Green" title="E40">A37</font></i></a>, <a class="txt" href="combgras.html#E24:20_1_2"><i><font color="Green" title="E41">A38</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">B</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E6:20_1_2_1"/><i><font color="Green" title="E44">A41</font></i>: 
<font color="Maroon" title="c10">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> <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"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>
<a NAME="E7:20_1_2_1"/>
( <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">S</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"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B</font> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">S</font> &amp; <font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E23:20_1_2"><i><font color="Green" title="E40">A37</font></i></a>, <a class="txt" href="combgras.html#E6:20_1_2_1"><i><font color="Green" title="E44">A41</font></i></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a>, <a class="ref" href="xboole_1.html#T10" title="XBOOLE_1:th.10">XBOOLE_1:10</a></i>;<br/>
<a NAME="E8:20_1_2_1"/><b>then </b><i><font color="Green" title="E45">A42</font></i>: 
( <font color="Maroon" title="c10">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">S</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="combgras.html#E23:20_1_2"><i><font color="Green" title="E40">A37</font></i></a>, <a class="txt" href="combgras.html#E6:20_1_2_1"><i><font color="Green" title="E44">A41</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E9:20_1_2_1"/>
for <font color="Olive" title="b1">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>  holds not <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A2</font>,<font color="Maroon" title="c10">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_2_1_1"/>
 ex <font color="Olive" title="b1">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A2</font>,<font color="Maroon" title="c10">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">L</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E3:20_1_2_1_1"/><i><font color="Green" title="E46">A43</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A2</font>,<font color="Maroon" title="c10">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c11">L</font>
 ;<br/>
<a NAME="E4:20_1_2_1_1"/>
<font color="Maroon" title="c11">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c12">L1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E6:20_1_2_1_1"/><i><font color="Green" title="E47">A44</font></i>: 
( <font color="Maroon" title="c11">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">L1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c12">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="combgras.html#E7:20"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/>
<a NAME="E7:20_1_2_1_1"/>
( <font color="Maroon" title="c7">A2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">L</font> &amp; <font color="Maroon" title="c10">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">L</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_1_2_1_1"><i><font color="Green" title="E46">A43</font></i></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></i>;<br/>
<a NAME="E8:20_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">L</font> &amp; <font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">L</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E4:20"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="combgras.html#E7:20_1_2"><i><font color="Green" title="E33">A30</font></i></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></i>;<br/>
<a NAME="E9:20_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">L</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E10:20_1_2_1_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">B</font></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c11">L</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E11:20_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c11">L</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E1:20"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="combgras.html#E3:20_1_2_1"><i><font color="Green" title="E43">A40</font></i></a>, <a class="txt" href="combgras.html#E8:20_1_2_1"><i><font color="Green" title="E45">A42</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:20_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="combgras.html#E8:20"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="combgras.html#E6:20_1_2_1_1"><i><font color="Green" title="E47">A44</font></i></a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></i>;<br/>
<b>hence </b><a NAME="E13:20_1_2_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:20_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="combgras.html#E1:20_1_2"><i><font color="Green" title="E30">A27</font></i></a>, <a class="txt" href="combgras.html#E4:20_1_2"><i><font color="Green" title="E31">A28</font></i></a>, <a class="txt" href="combgras.html#E7:20_1_2"><i><font color="Green" title="E33">A30</font></i></a>, <a class="txt" href="combgras.html#E8:20_1_2_1"><i><font color="Green" title="E45">A42</font></i></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:20_1_2"/>
not <font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_2_2"/>
<font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:20_1_2_2"/><i><font color="Green" title="E43">A45</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:20_1_2_2"/>
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">S</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_1_2_2"><i><font color="Green" title="E43">A45</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:20_1_2_2"/><b>then </b><i><font color="Green" title="E44">A46</font></i>: 
( not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> &amp; not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E6:20_1_2_2"/><b>then </b><i><font color="Green" title="E45">A47</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 &amp; <font color="Maroon" title="c4">S</font> <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"><font color="Maroon" title="c10">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="combgras.html#E4:20_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E7:20_1_2_2"/><b>then </b>
<font color="Maroon" title="c4">S</font> <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"><font color="Maroon" title="c10">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="combgras.html#E6:20"><i><font color="Green" title="E23">A5</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">B</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E9:20_1_2_2"/><i><font color="Green" title="E46">A48</font></i>: 
<font color="Maroon" title="c11">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> <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"><font color="Maroon" title="c10">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>
<a NAME="E10:20_1_2_2"/>
( <font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">B</font> &amp; <font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E5:20_1_2_2"><i><font color="Green" title="E44">A46</font></i></a>, <a class="txt" href="combgras.html#E9:20_1_2_2"><i><font color="Green" title="E46">A48</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E11:20_1_2_2"/><b>then </b><i><font color="Green" title="E47">A49</font></i>: 
<font color="Maroon" title="c11">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="combgras.html#E6:20_1_2_2"><i><font color="Green" title="E45">A47</font></i></a>, <a class="txt" href="combgras.html#E9:20_1_2_2"><i><font color="Green" title="E46">A48</font></i></a></i>;<br/>
<a NAME="E12:20_1_2_2"/>
<font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:20_1_2_2_1"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">B</font>
 ;<br/>

<a NAME="E2:20_1_2_2_1"/><b>then </b><i><font color="Green" title="E48">A50</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> <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"><font color="Maroon" title="c10">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="combgras.html#E9:20_1_2_2"><i><font color="Green" title="E46">A48</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:20_1_2_2_1"/><b>then </b>
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> or <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:20_1_2_2_1"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font>
 <b>by </b><i><a class="txt" href="combgras.html#E5:20_1_2_2"><i><font color="Green" title="E44">A46</font></i></a>, <a class="txt" href="combgras.html#E2:20_1_2_2_1"><i><font color="Green" title="E48">A50</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:20_1_2_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">B</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E14:20_1_2_2"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2 &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )
 
<b>by </b><i><a class="txt" href="combgras.html#E2:20"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="combgras.html#E16:20_1_2"><i><font color="Green" title="E38">A35</font></i></a>, <a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E15:20_1_2_2"/><b>then </b><i><font color="Green" title="E48">A51</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="txt" href="combgras.html#E2:20"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></i>;<br/>
<a NAME="E16:20_1_2_2"/>
for <font color="Olive" title="b1">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>  holds not <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A2</font>,<font color="Maroon" title="c11">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_2_2_2"/>
 ex <font color="Olive" title="b1">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A2</font>,<font color="Maroon" title="c11">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">L</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E3:20_1_2_2_2"/><i><font color="Green" title="E49">A52</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A2</font>,<font color="Maroon" title="c11">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c12">L</font>
 ;<br/>
<a NAME="E4:20_1_2_2_2"/>
<font color="Maroon" title="c12">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c13">L1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E6:20_1_2_2_2"/><i><font color="Green" title="E50">A53</font></i>: 
( <font color="Maroon" title="c12">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">L1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c13">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="combgras.html#E7:20"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/>
<a NAME="E7:20_1_2_2_2"/>
( <font color="Maroon" title="c7">A2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">L</font> &amp; <font color="Maroon" title="c11">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">L</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_1_2_2_2"><i><font color="Green" title="E49">A52</font></i></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></i>;<br/>
<a NAME="E8:20_1_2_2_2"/><b>then </b>
( <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">L</font> &amp; <font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">L</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E4:20"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="combgras.html#E7:20_1_2"><i><font color="Green" title="E33">A30</font></i></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></i>;<br/>
<a NAME="E9:20_1_2_2_2"/><b>then </b><i><font color="Green" title="E51">A54</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">L</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E10:20_1_2_2_2"/>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_2_2_2_1"/>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">B</font>
 ;<br/>

<a NAME="E2:20_1_2_2_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E9:20_1_2_2"><i><font color="Green" title="E46">A48</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E3:20_1_2_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="combgras.html#E5:20_1_2_2"><i><font color="Green" title="E44">A46</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:20_1_2_2_2"/><b>then </b>
( <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">B</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">B</font></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="combgras.html#E9:20_1_2"><i><font color="Green" title="E34">A31</font></i></a>, <a class="txt" href="combgras.html#E6:20_1_2_2"><i><font color="Green" title="E45">A47</font></i></a>, <a class="txt" href="combgras.html#E9:20_1_2_2"><i><font color="Green" title="E46">A48</font></i></a>, <a class="txt" href="combgras.html#E6:20_1_2_2_2"><i><font color="Green" title="E50">A53</font></i></a>, <a class="txt" href="combgras.html#E9:20_1_2_2_2"><i><font color="Green" title="E51">A54</font></i></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E12:20_1_2_2_2"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c11">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="combgras.html#E4:20_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="combgras.html#E9:20_1_2"><i><font color="Green" title="E34">A31</font></i></a>, <a class="txt" href="combgras.html#E5:20_1_2_2"><i><font color="Green" title="E44">A46</font></i></a>, <a class="txt" href="combgras.html#E9:20_1_2_2"><i><font color="Green" title="E46">A48</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E13:20_1_2_2_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="txt" href="combgras.html#E5:20"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></i>;<br/>
<b>hence </b><a NAME="E14:20_1_2_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="combgras.html#E15:20_1_2_2"><i><font color="Green" title="E48">A51</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:20_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="combgras.html#E1:20_1_2"><i><font color="Green" title="E30">A27</font></i></a>, <a class="txt" href="combgras.html#E4:20_1_2"><i><font color="Green" title="E31">A28</font></i></a>, <a class="txt" href="combgras.html#E7:20_1_2"><i><font color="Green" title="E33">A30</font></i></a>, <a class="txt" href="combgras.html#E11:20_1_2_2"><i><font color="Green" title="E47">A49</font></i></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E27:20_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="combgras.html#E25:20_1_2"><i><font color="Green" title="E42">A39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:20_1"/>
<font color="Maroon" title="c3">K</font> is <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a>
 <b>by </b><i><a class="txt" href="combgras.html#E5:20_1"><i><font color="Green" title="E29">A11</font></i></a>, <a class="ref" href="combgras.html#D3" target="_self" title="COMBGRAS:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:20"/>
( <font color="Maroon" title="c3">K</font> is <a href="combgras.html#V4" title="COMBGRAS:attr.4">TOP</a> implies <font color="Maroon" title="c3">K</font> is <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2"/>
<font color="Maroon" title="c3">K</font> is <a href="combgras.html#V4" title="COMBGRAS:attr.4">TOP</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">S</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E3:20_2"/><i><font color="Green" title="E27">A55</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> : (  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="combgras.html#D5" target="_self" title="COMBGRAS:def.5">Def5</a></i>;<br/>
<a NAME="E4:20_2"/><i><font color="Green" title="E28">A56</font></i>: 
<font color="Maroon" title="c4">S</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="combgras.html#E8:20"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="combgras.html#E3:20_2"><i><font color="Green" title="E27">A55</font></i></a>, <a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a></i>;<br/>
<a NAME="E5:20_2"/><i><font color="Green" title="E29">A57</font></i>: 
<font color="Maroon" title="c3">K</font> is <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">COMBGRAS:def 2</a><br/>

<b>assume </b><a NAME="E1:20_2_1"/><i><font color="Green" title="E30">A58</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">A1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E3:20_2_1"/><i><font color="Green" title="E31">A59</font></i>: 
( <font color="Maroon" title="c7">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c7">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c7">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font> )
 <b>by </b><i><a class="txt" href="combgras.html#E3:20_2"><i><font color="Green" title="E27">A55</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">B1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E5:20_2_1"/><i><font color="Green" title="E32">A60</font></i>: 
( <font color="Maroon" title="c8">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c8">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c8">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font> )
 <b>by </b><i><a class="txt" href="combgras.html#E3:20_2"><i><font color="Green" title="E27">A55</font></i></a>, <a class="txt" href="combgras.html#E1:20_2_1"><i><font color="Green" title="E30">A58</font></i></a></i>;<br/>
<a NAME="E6:20_2_1"/>
<font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="combgras.html#E7:20"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="combgras.html#E3:20_2"><i><font color="Green" title="E27">A55</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">L</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E8:20_2_1"/><i><font color="Green" title="E33">A61</font></i>: 
<font color="Maroon" title="c9">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font>
 ;<br/>
<a NAME="E9:20_2_1"/>
( <font color="Maroon" title="c5">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L</font> &amp; <font color="Maroon" title="c6">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E4:20"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="combgras.html#E3:20_2_1"><i><font color="Green" title="E31">A59</font></i></a>, <a class="txt" href="combgras.html#E5:20_2_1"><i><font color="Green" title="E32">A60</font></i></a>, <a class="txt" href="combgras.html#E8:20_2_1"><i><font color="Green" title="E33">A61</font></i></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></i>;<br/>
<a NAME="E10:20_2_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c9">L</font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></i>;<br/>
<b>hence </b><a NAME="E11:20_2_1"/>
 ex <font color="Olive" title="b1">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:20_2"/>
for <font color="Olive" title="b1">U</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">U</font> is <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> &amp; <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U</font> holds <br/><font color="Olive" title="b1">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">U</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:20_2_2"/><i><font color="Green" title="E30">A62</font></i>: 
( <font color="Maroon" title="c5">U</font> is <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> &amp; <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">U</font> &amp; <font color="Maroon" title="c5">U</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">K</font> )
 ;<br/>

<a NAME="E2:20_2_2"/><b>then </b>
not <font color="Maroon" title="c5">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:20_2_2"/><i><font color="Green" title="E31">A63</font></i>: 
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">U</font> &amp; not <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:20_2_2"/><i><font color="Green" title="E32">A64</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="combgras.html#E4:20_2_2"><i><font color="Green" title="E31">A63</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">A2</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E7:20_2_2"/><i><font color="Green" title="E33">A65</font></i>: 
<font color="Maroon" title="c7">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font>
 <b>by </b><i><a class="txt" href="combgras.html#E4:20_2_2"><i><font color="Green" title="E31">A63</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">A1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E9:20_2_2"/><i><font color="Green" title="E34">A66</font></i>: 
( <font color="Maroon" title="c8">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c8">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 <b>by </b><i><a class="txt" href="combgras.html#E6:20"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="combgras.html#E5:20_2_2"><i><font color="Green" title="E32">A64</font></i></a></i>;<br/>
<a NAME="E10:20_2_2"/><i><font color="Green" title="E35">A67</font></i>: 
( not <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font> &amp; <font color="Maroon" title="c6">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E8:20"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="combgras.html#E3:20_2"><i><font color="Green" title="E27">A55</font></i></a>, <a class="txt" href="combgras.html#E4:20_2_2"><i><font color="Green" title="E31">A63</font></i></a>, <a class="txt" href="combgras.html#E9:20_2_2"><i><font color="Green" title="E34">A66</font></i></a>, <a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:20_2_2"/><i><font color="Green" title="E36">A68</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> &amp; not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E13:20_2_2"/>
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E14:20_2_2"/><b>then </b><i><font color="Green" title="E37">A69</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">A</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">S</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_2"><i><font color="Green" title="E27">A55</font></i></a>, <a class="txt" href="combgras.html#E9:20_2_2"><i><font color="Green" title="E34">A66</font></i></a>, <a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/>
<a NAME="E15:20_2_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">S</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_2"><i><font color="Green" title="E27">A55</font></i></a>, <a class="txt" href="combgras.html#E9:20_2_2"><i><font color="Green" title="E34">A66</font></i></a></i>;<br/>
<a NAME="E16:20_2_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">A</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">S</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E14:20_2_2"><i><font color="Green" title="E37">A69</font></i></a>, <a class="ref" href="card_1.html#T13" title="CARD_1:th.13">CARD_1:13</a></i>;<br/>
<a NAME="E17:20_2_2"/><b>then </b>
<font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T4" title="CARD_2:th.4">CARD_2:4</a></i>;<br/>
<a NAME="E18:20_2_2"/><b>then </b><i><font color="Green" title="E38">A70</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="card_4.html#T17" title="CARD_4:th.17">CARD_4:17</a></i>;<br/>
<a NAME="E19:20_2_2"/>
2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2_2_1"/>
not 2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font></span>)</span>
 ;<br/>

<a NAME="E2:20_2_2_1"/><b>then </b><i><font color="Green" title="E39">A71</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a></i>;<br/>
<a NAME="E3:20_2_2_1"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 or  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T13" title="ORDINAL1:th.13">ORDINAL1:13</a></i>;<br/>
<a NAME="E4:20_2_2_1"/><b>then </b><i><font color="Green" title="E40">A72</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="combgras.html#E8:20"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></i>;<br/>
<a NAME="E5:20_2_2_1"/>
not  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2_2_1_1"/><i><font color="Green" title="E41">A73</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<a NAME="E2:20_2_2_1_1"/><i><font color="Green" title="E42">A74</font></i>: 
<font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E3:20_2_2_1_1"/>
<font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="combgras.html#E4:20_2"><i><font color="Green" title="E28">A56</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E4:20_2_2_1_1"/><b>then </b><i><font color="Green" title="E43">A75</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font> &amp; <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_2"><i><font color="Green" title="E27">A55</font></i></a>, <a class="txt" href="combgras.html#E4:20_2"><i><font color="Green" title="E28">A56</font></i></a>, <a class="txt" href="combgras.html#E1:20_2_2_1_1"><i><font color="Green" title="E41">A73</font></i></a>, <a class="txt" href="combgras.html#E2:20_2_2_1_1"><i><font color="Green" title="E42">A74</font></i></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a>, <a class="ref" href="xboole_1.html#T48" title="XBOOLE_1:th.48">XBOOLE_1:48</a></i>;<br/>
<a NAME="E5:20_2_2_1_1"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="txt" href="combgras.html#E8:20"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a></i>;<br/>
<b>hence </b><a NAME="E6:20_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="combgras.html#E9:20_2_2"><i><font color="Green" title="E34">A66</font></i></a>, <a class="txt" href="combgras.html#E10:20_2_2"><i><font color="Green" title="E35">A67</font></i></a>, <a class="txt" href="combgras.html#E4:20_2_2_1_1"><i><font color="Green" title="E43">A75</font></i></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:20_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="combgras.html#E18:20_2_2"><i><font color="Green" title="E38">A70</font></i></a>, <a class="txt" href="combgras.html#E2:20_2_2_1"><i><font color="Green" title="E39">A71</font></i></a>, <a class="txt" href="combgras.html#E4:20_2_2_1"><i><font color="Green" title="E40">A72</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c10">B1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E21:20_2_2"/><i><font color="Green" title="E39">A76</font></i>: 
( <font color="Maroon" title="c10">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c10">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></i>;<br/>
<a NAME="E22:20_2_2"/><i><font color="Green" title="E40">A77</font></i>: 
( <font color="Maroon" title="c10">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font> &amp; <font color="Maroon" title="c10">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">A</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E21:20_2_2"><i><font color="Green" title="E39">A76</font></i></a>, <a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></i>;<br/>
<a NAME="E23:20_2_2"/><b>then </b><i><font color="Green" title="E41">A78</font></i>: 
<font color="Maroon" title="c10">B1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="combgras.html#E4:20_2"><i><font color="Green" title="E28">A56</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<a NAME="E24:20_2_2"/><b>then </b><i><font color="Green" title="E42">A79</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_2"><i><font color="Green" title="E27">A55</font></i></a>, <a class="txt" href="combgras.html#E4:20_2"><i><font color="Green" title="E28">A56</font></i></a>, <a class="txt" href="combgras.html#E21:20_2_2"><i><font color="Green" title="E39">A76</font></i></a>, <a class="txt" href="combgras.html#E22:20_2_2"><i><font color="Green" title="E40">A77</font></i></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>;<br/>
<a NAME="E25:20_2_2"/>
<font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="txt" href="combgras.html#E2:20"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E26:20_2_2"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">B1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="combgras.html#E2:20"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="combgras.html#E5:20"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="combgras.html#E24:20_2_2"><i><font color="Green" title="E42">A79</font></i></a>, <a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">B2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E28:20_2_2"/><i><font color="Green" title="E43">A80</font></i>: 
( <font color="Maroon" title="c11">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">B1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c11">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2 )
 <b>by </b><i><a class="txt" href="combgras.html#E2:20"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></i>;<br/>
<a NAME="E29:20_2_2"/><i><font color="Green" title="E44">A81</font></i>: 
( <font color="Maroon" title="c11">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font> &amp; <font color="Maroon" title="c11">B2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">B1</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E28:20_2_2"><i><font color="Green" title="E43">A80</font></i></a>, <a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></i>;<br/>
<a NAME="E30:20_2_2"/><b>then </b>
( <font color="Maroon" title="c11">B2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c10">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c11">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E4:20_2"><i><font color="Green" title="E28">A56</font></i></a>, <a class="txt" href="combgras.html#E21:20_2_2"><i><font color="Green" title="E39">A76</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E31:20_2_2"/><b>then </b><i><font color="Green" title="E45">A82</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">B2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> &amp; <font color="Maroon" title="c10">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c10">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E21:20_2_2"><i><font color="Green" title="E39">A76</font></i></a>, <a class="txt" href="combgras.html#E22:20_2_2"><i><font color="Green" title="E40">A77</font></i></a>, <a class="txt" href="combgras.html#E23:20_2_2"><i><font color="Green" title="E41">A78</font></i></a>, <a class="txt" href="combgras.html#E28:20_2_2"><i><font color="Green" title="E43">A80</font></i></a>, <a class="txt" href="combgras.html#E29:20_2_2"><i><font color="Green" title="E44">A81</font></i></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E32:20_2_2"/><b>then </b>
<font color="Maroon" title="c10">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="combgras.html#E6:20"><i><font color="Green" title="E23">A5</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">B</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E34:20_2_2"/><i><font color="Green" title="E46">A83</font></i>: 
<font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">B2</font>
 ;<br/>
<a NAME="E35:20_2_2"/><i><font color="Green" title="E47">A84</font></i>: 
( <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font> &amp; not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B1</font> &amp; not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B2</font> &amp; <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">B2</font></span>)</span> &amp; <font color="Maroon" title="c12">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E12:20_2_2"><i><font color="Green" title="E36">A68</font></i></a>, <a class="txt" href="combgras.html#E22:20_2_2"><i><font color="Green" title="E40">A77</font></i></a>, <a class="txt" href="combgras.html#E29:20_2_2"><i><font color="Green" title="E44">A81</font></i></a>, <a class="txt" href="combgras.html#E34:20_2_2"><i><font color="Green" title="E46">A83</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a>, <a class="ref" href="xboole_1.html#T29" title="XBOOLE_1:th.29">XBOOLE_1:29</a></i>;<br/>
<a NAME="E36:20_2_2"/><i><font color="Green" title="E48">A85</font></i>: 
( not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">B2</font> &amp; <font color="Maroon" title="c10">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c12">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_2"><i><font color="Green" title="E27">A55</font></i></a>, <a class="txt" href="combgras.html#E12:20_2_2"><i><font color="Green" title="E36">A68</font></i></a>, <a class="txt" href="combgras.html#E22:20_2_2"><i><font color="Green" title="E40">A77</font></i></a>, <a class="txt" href="combgras.html#E31:20_2_2"><i><font color="Green" title="E45">A82</font></i></a>, <a class="txt" href="combgras.html#E34:20_2_2"><i><font color="Green" title="E46">A83</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E37:20_2_2"/><i><font color="Green" title="E49">A86</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2_2_2"/>
not <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font></span>)</span>
 ;<br/>

<a NAME="E2:20_2_2_2"/><b>then </b><i><font color="Green" title="E50">A87</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="combgras.html#E8:20"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a></i>;<br/>
<a NAME="E3:20_2_2_2"/><i><font color="Green" title="E51">A88</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2_2_2_1"/>
not <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</font>
 ;<br/>

<a NAME="E2:20_2_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">B1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:20_2_2_2_1"/><i><font color="Green" title="E52">A89</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">B1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:20_2_2_2_1"/>
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">B1</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E4:20_2_2_2_1"><i><font color="Green" title="E52">A89</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:20_2_2_2_1"/><b>then </b>
( ( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B1</font> ) &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:20_2_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="combgras.html#E34:20_2_2"><i><font color="Green" title="E46">A83</font></i></a>, <a class="txt" href="combgras.html#E36:20_2_2"><i><font color="Green" title="E48">A85</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:20_2_2_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c10">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E12:20_2_2"><i><font color="Green" title="E36">A68</font></i></a>, <a class="txt" href="combgras.html#E36:20_2_2"><i><font color="Green" title="E48">A85</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E5:20_2_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c10">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c10">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">B</font> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E34:20_2_2"><i><font color="Green" title="E46">A83</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a>, <a class="ref" href="xboole_1.html#T10" title="XBOOLE_1:th.10">XBOOLE_1:10</a></i>;<br/>
<a NAME="E6:20_2_2_2"/><b>then </b>
( <font color="Maroon" title="c10">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c10">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</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"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E35:20_2_2"><i><font color="Green" title="E47">A84</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a>, <a class="ref" href="xboole_1.html#T10" title="XBOOLE_1:th.10">XBOOLE_1:10</a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>;<br/>
<a NAME="E7:20_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E8:20_2_2_2"/><b>then </b><i><font color="Green" title="E52">A90</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">B1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="combgras.html#E2:20_2_2_2"><i><font color="Green" title="E50">A87</font></i></a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E9:20_2_2_2"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or (  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font> ) )
 
<b>by </b><i><a class="txt" href="combgras.html#E8:20"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="combgras.html#E2:20_2_2_2"><i><font color="Green" title="E50">A87</font></i></a>, <a class="ref" href="ordinal1.html#T13" title="ORDINAL1:th.13">ORDINAL1:13</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E10:20_2_2_2"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or (  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="combgras.html#E9:20_2_2"><i><font color="Green" title="E34">A66</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></i>;<br/>
<a NAME="E11:20_2_2_2"/><b>then </b>
( (  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> or  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> ) &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )
 
<b>by </b><i><a class="txt" href="combgras.html#E9:20_2_2"><i><font color="Green" title="E34">A66</font></i></a>, <a class="txt" href="combgras.html#E31:20_2_2"><i><font color="Green" title="E45">A82</font></i></a>, <a class="txt" href="combgras.html#E34:20_2_2"><i><font color="Green" title="E46">A83</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E12:20_2_2_2"/><b>then </b><i><font color="Green" title="E53">A91</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 or  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E5:20"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></i>;<br/>
<a NAME="E13:20_2_2_2"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">B1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="txt" href="combgras.html#E10:20_2_2"><i><font color="Green" title="E35">A67</font></i></a>, <a class="txt" href="combgras.html#E21:20_2_2"><i><font color="Green" title="E39">A76</font></i></a>, <a class="txt" href="combgras.html#E23:20_2_2"><i><font color="Green" title="E41">A78</font></i></a>, <a class="txt" href="combgras.html#E35:20_2_2"><i><font color="Green" title="E47">A84</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a>, <a class="ref" href="finset_1.html#T14" title="FINSET_1:th.14">FINSET_1:14</a>, <a class="ref" href="finset_1.html#T15" title="FINSET_1:th.15">FINSET_1:15</a></i>;<br/>
<a NAME="E14:20_2_2_2"/><b>then </b>
( (  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">B1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 3 or  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">B1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 ) &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">B1</font></span>)</span></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_2_2_2"><i><font color="Green" title="E51">A88</font></i></a>, <a class="txt" href="combgras.html#E8:20_2_2_2"><i><font color="Green" title="E52">A90</font></i></a>, <a class="txt" href="combgras.html#E12:20_2_2_2"><i><font color="Green" title="E53">A91</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></i>;<br/>
<a NAME="E15:20_2_2_2"/><b>then </b>
( <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="combgras.html#E8:20_2_2_2"><i><font color="Green" title="E52">A90</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E16:20_2_2_2"/><b>then </b>
( <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="combgras.html#E8:20"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></i>;<br/>
<b>hence </b><a NAME="E17:20_2_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E38:20_2_2"/>
for <font color="Olive" title="b1">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>  holds not <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A2</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2_2_3"/>
 ex <font color="Olive" title="b1">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> st <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A2</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">L</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E3:20_2_2_3"/><i><font color="Green" title="E50">A92</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A2</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c13">L</font>
 ;<br/>
<a NAME="E4:20_2_2_3"/>
<font color="Maroon" title="c13">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c14">L1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E6:20_2_2_3"/><i><font color="Green" title="E51">A93</font></i>: 
( <font color="Maroon" title="c13">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">L1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c14">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="combgras.html#E7:20"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/>
<a NAME="E7:20_2_2_3"/>
( <font color="Maroon" title="c7">A2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">L</font> &amp; <font color="Maroon" title="c12">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">L</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:20_2_2_3"><i><font color="Green" title="E50">A92</font></i></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></i>;<br/>
<a NAME="E8:20_2_2_3"/><b>then </b>
( <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">L</font> &amp; <font color="Maroon" title="c12">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">L</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E4:20"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="combgras.html#E7:20_2_2"><i><font color="Green" title="E33">A65</font></i></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></i>;<br/>
<a NAME="E9:20_2_2_3"/><b>then </b>
<font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">L</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E10:20_2_2_3"/><b>then </b>
( <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">B</font></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="combgras.html#E9:20_2_2"><i><font color="Green" title="E34">A66</font></i></a>, <a class="txt" href="combgras.html#E12:20_2_2"><i><font color="Green" title="E36">A68</font></i></a>, <a class="txt" href="combgras.html#E31:20_2_2"><i><font color="Green" title="E45">A82</font></i></a>, <a class="txt" href="combgras.html#E34:20_2_2"><i><font color="Green" title="E46">A83</font></i></a>, <a class="txt" href="combgras.html#E36:20_2_2"><i><font color="Green" title="E48">A85</font></i></a>, <a class="txt" href="combgras.html#E6:20_2_2_3"><i><font color="Green" title="E51">A93</font></i></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E11:20_2_2_3"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="combgras.html#E37:20_2_2"><i><font color="Green" title="E49">A86</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E12:20_2_2_3"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="combgras.html#E8:20"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></i>;<br/>
<b>hence </b><a NAME="E13:20_2_2_3"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E39:20_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="combgras.html#E1:20_2_2"><i><font color="Green" title="E30">A62</font></i></a>, <a class="txt" href="combgras.html#E4:20_2_2"><i><font color="Green" title="E31">A63</font></i></a>, <a class="txt" href="combgras.html#E7:20_2_2"><i><font color="Green" title="E33">A65</font></i></a>, <a class="txt" href="combgras.html#E36:20_2_2"><i><font color="Green" title="E48">A85</font></i></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:20_2"/>
<font color="Maroon" title="c3">K</font> is <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a>
 <b>by </b><i><a class="txt" href="combgras.html#E5:20_2"><i><font color="Green" title="E29">A57</font></i></a>, <a class="ref" href="combgras.html#D3" target="_self" title="COMBGRAS:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:20"/>
( ( <font color="Maroon" title="c3">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> or <font color="Maroon" title="c3">K</font> is <a href="combgras.html#V4" title="COMBGRAS:attr.4">TOP</a> ) implies <font color="Maroon" title="c3">K</font> is <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a> )
 <b>by </b><i><a class="txt" href="combgras.html#E9:20"><i><font color="Green" title="E26">A8</font></i></a></i>;<br/>


</div>
