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

<a NAME="E1:21"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
 <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
 
;<br/>
<a NAME="E2:21"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
 <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
 
;<br/>
<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being   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>   st 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="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">X</font> holds <br/> for <font color="Olive" title="b2">K</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="Olive" title="b1">X</font>)</span>)</span>  holds <br/>(  not <font color="Olive" title="b2">K</font> is  <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a>  or <font color="Olive" title="b2">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  or <font color="Olive" title="b2">K</font> is  <a href="combgras.html#V4" title="COMBGRAS:attr.4">TOP</a>  )</span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 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="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="c2">X</font> implies  for <font color="Olive" title="b1">K</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>  holds <br/>(  not <font color="Olive" title="b1">K</font> is  <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a>  or <font color="Olive" title="b1">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  or <font color="Olive" title="b1">K</font> is  <a href="combgras.html#V4" title="COMBGRAS:attr.4">TOP</a>  ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:21"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">and </span><br/><a NAME="E4:21"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<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="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">K</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>  holds <br/>(  not <font color="Olive" title="b1">K</font> is  <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a>  or <font color="Olive" title="b1">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  or <font color="Olive" title="b1">K</font> is  <a href="combgras.html#V4" title="COMBGRAS:attr.4">TOP</a>  )</span><br/>

<a NAME="E5:21"/>
<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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E6:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<a NAME="E7:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<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>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E8:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A7</font></span>: 
 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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></span>;<br/>
<a NAME="E9:21"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
 <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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T38" title="NAT_1:th.38">NAT_1:38</a></span>;<br/>
<a NAME="E10:21"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">K</font> is  <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a>  or <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>  )</span><br/>

<a NAME="E11:21"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
 <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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T38" title="NAT_1:th.38">NAT_1:38</a></span>;<br/>
<a NAME="E12:21"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c3">K</font>
 
;<br/>
<a NAME="E13:21"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> <font color="Maroon" title="c3">K</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>;<br/>
<a NAME="E14:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A11</font></span>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T8" title="ORDINAL1:th.8">ORDINAL1:8</a></span>;<br/>
<span class="kw">assume </span><a NAME="E15:21"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
<font color="Maroon" title="c3">K</font> is  <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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>  )</span><br/>

<a NAME="E16:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A13</font></span>: 
<font color="Maroon" title="c3">K</font> is  <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="combgras.html#D3" target="_self" title="COMBGRAS:def.3">Def3</a></span>;<br/>
<a NAME="E17:21"/><span class="lab"><font color="Green" title="E33">A14</font></span>: 
 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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></span>;<br/>
<a NAME="E18:21"/>
<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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E19:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A15</font></span>: 
<font color="Maroon" title="c1">k</font> <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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<a NAME="E20:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A16</font></span>: 
<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> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E21:21"/>
<font color="Maroon" title="c3">K</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c4">A1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:21_1"/><span class="lab"><font color="Green" title="E36">A17</font></span>: 
<font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E3:21_1"/><span class="lab"><font color="Green" title="E37">A18</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:21"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></span>;<br/>
<a NAME="E4:21_1"/>
<font color="Maroon" title="c4">A1</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E2:21_1"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E37">A18</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">A</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><span class="kw"> such that </span><br/><a NAME="E6:21_1"/><span class="lab"><font color="Green" title="E38">A19</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A1</font>
 ;<br/>
<a NAME="E7:21_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#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="21_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_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>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="txt" href="#E6:21_1"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E3:21_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:21_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:21_1"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:21"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="txt" href="#E6:21_1"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="ref" href="card_1.html#T3" title="CARD_1:th.3">CARD_1:3</a></span>;<br/>
<a NAME="E9:21_1"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T68" title="CARD_1:th.68">CARD_1:68</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:21_1"/><span class="lab"><font color="Green" title="E39">A20</font></span>: 
<font color="Maroon" title="c6">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E12:21_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21_1"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<a NAME="E13:21_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A21</font></span>: 
<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="c6">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E6:21_1"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E14:21_1"/><span class="lab"><font color="Green" title="E41">A22</font></span>: 
 not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21_1"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E15:21_1"/>
<font color="Maroon" title="c5">A</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="txt" href="#E6:21_1"><span class="lab"><font color="Green" title="E38">A19</font></span></a></span>;<br/>
<a NAME="E16:21_1"/><span class="kw">then </span>
 <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="c6">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="txt" href="#E6:21_1"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="txt" href="#E14:21_1"><span class="lab"><font color="Green" title="E41">A22</font></span></a>, <a class="ref" href="card_2.html#T41" title="CARD_2:th.41">CARD_2:41</a></span>;<br/>
<a NAME="E17:21_1"/><span class="kw">then </span>
<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="c6">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="txt" href="#E13:21_1"><span class="lab"><font color="Green" title="E40">A21</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">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><span class="kw"> such that </span><br/><a NAME="E19:21_1"/><span class="lab"><font color="Green" title="E42">A23</font></span>: 
<font color="Maroon" title="c7">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="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">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><span class="kw"> such that </span><br/><a NAME="E21:21_1"/><span class="lab"><font color="Green" title="E43">A24</font></span>: 
<font color="Maroon" title="c8">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>
<a NAME="E22:21_1"/>
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:21_1"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E23:21_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A25</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E24:21_1"/><span class="lab"><font color="Green" title="E45">A26</font></span>: 
<font color="Maroon" title="c8">U</font> is  <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">B</font>, <font color="Maroon" title="c10">C</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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">COMBGRAS:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">U</font> &amp; <font color="Maroon" title="c10">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">U</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="c9">B</font>,<font color="Maroon" title="c10">C</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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_1_2"/>
( <font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">U</font> &amp; <font color="Maroon" title="c10">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">U</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c9">B</font>,<font color="Maroon" title="c10">C</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></span><br/>

<a NAME="E2:21_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c9">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">L</font> &amp; <font color="Maroon" title="c10">C</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:21_1"><span class="lab"><font color="Green" title="E44">A25</font></span></a>, <a class="txt" href="#E21:21_1"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E3:21_1_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c10">C</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="c7">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:21_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="c9">B</font>,<font color="Maroon" title="c10">C</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E25:21_1"/><span class="lab"><font color="Green" title="E46">A27</font></span>: 
<font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E26:21_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:21_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E25:21_1"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E21:21_1"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="txt" href="#E24:21_1"><span class="lab"><font color="Green" title="E45">A26</font></span></a>, <a class="ref" href="combgras.html#D3" target="_self" title="COMBGRAS:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E22:21"/><span class="kw">then </span>
1 <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="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/>
<a NAME="E23:21"/><span class="kw">then </span>
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> <font color="Maroon" title="c3">K</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>;<br/>
<a NAME="E24:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A28</font></span>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or 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> <font color="Maroon" title="c3">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T8" title="ORDINAL1:th.8">ORDINAL1:8</a></span>;<br/>
<a NAME="E25:21"/><span class="lab"><font color="Green" title="E37">A29</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E26:21"/><span class="lab"><font color="Green" title="E38">A30</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">A3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:21_2"/><span class="lab"><font color="Green" title="E39">A31</font></span>: 
<font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">A3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T42" title="CARD_2:th.42">CARD_2:42</a></span>;<br/>
<a NAME="E4:21_2"/><span class="lab"><font color="Green" title="E40">A32</font></span>: 
<font color="Maroon" title="c4">A3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_2"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">A</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><span class="kw"> such that </span><br/><a NAME="E6:21_2"/><span class="lab"><font color="Green" title="E41">A33</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A3</font>
 ;<br/>
<a NAME="E7:21_2"/>
<font color="Maroon" title="c4">A3</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_2"><span class="lab"><font color="Green" title="E40">A32</font></span></a></span>;<br/>
<a NAME="E8:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A34</font></span>: 
 ex <font color="Olive" title="b1">A4</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A4</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">A4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E6:21_2"><span class="lab"><font color="Green" title="E41">A33</font></span></a></span>;<br/>
<a NAME="E9:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A35</font></span>: 
<font color="Maroon" title="c5">A</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<a NAME="E10:21_2"/><span class="lab"><font color="Green" title="E44">A36</font></span>: 
 <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="21_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_2_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E8:21_2"><span class="lab"><font color="Green" title="E42">A34</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E3:21_2_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:21_2_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:21_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">A</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> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E19:21"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="txt" href="#E8:21_2"><span class="lab"><font color="Green" title="E42">A34</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E12:21_2"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_2"><span class="lab"><font color="Green" title="E44">A36</font></span></a>, <a class="ref" href="card_1.html#T3" title="CARD_1:th.3">CARD_1:3</a></span>;<br/>
<a NAME="E13:21_2"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T68" title="CARD_1:th.68">CARD_1:68</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E15:21_2"/><span class="lab"><font color="Green" title="E45">A37</font></span>: 
<font color="Maroon" title="c6">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E16:21_2"/><span class="lab"><font color="Green" title="E46">A38</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21_2"><span class="lab"><font color="Green" title="E45">A37</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<a NAME="E17:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A39</font></span>: 
<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="c6">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_2"><span class="lab"><font color="Green" title="E42">A34</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E18:21_2"/>
 not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21_2"><span class="lab"><font color="Green" title="E45">A37</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E19:21_2"/><span class="kw">then </span>
 <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="c6">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_2"><span class="lab"><font color="Green" title="E42">A34</font></span></a>, <a class="txt" href="#E9:21_2"><span class="lab"><font color="Green" title="E43">A35</font></span></a>, <a class="ref" href="card_2.html#T41" title="CARD_2:th.41">CARD_2:41</a></span>;<br/>
<a NAME="E20:21_2"/><span class="kw">then </span>
<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="c6">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="txt" href="#E17:21_2"><span class="lab"><font color="Green" title="E47">A39</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">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><span class="kw"> such that </span><br/><a NAME="E22:21_2"/><span class="lab"><font color="Green" title="E48">A40</font></span>: 
<font color="Maroon" title="c7">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="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>
<a NAME="E23:21_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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E24:21_2"/><span class="kw">then </span>
<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> <font color="Maroon" title="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E8:21_2"><span class="lab"><font color="Green" title="E42">A34</font></span></a>, <a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">B2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E26:21_2"/><span class="lab"><font color="Green" title="E49">A41</font></span>: 
<font color="Maroon" title="c8">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">and </span><br/><a NAME="E27:21_2"/><span class="lab"><font color="Green" title="E50">A42</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c8">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> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></span>;<br/>
<a NAME="E28:21_2"/><span class="lab"><font color="Green" title="E51">A43</font></span>: 
<font color="Maroon" title="c8">B2</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E27:21_2"><span class="lab"><font color="Green" title="E50">A42</font></span></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E29:21_2"/>
<font color="Maroon" title="c8">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_2"><span class="lab"><font color="Green" title="E42">A34</font></span></a>, <a class="txt" href="#E26:21_2"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E30:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A44</font></span>: 
<font color="Maroon" title="c8">B2</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="c6">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E31:21_2"/>
 not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21_2"><span class="lab"><font color="Green" title="E45">A37</font></span></a>, <a class="txt" href="#E26:21_2"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E32:21_2"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B2</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="c6">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:21_2"><span class="lab"><font color="Green" title="E50">A42</font></span></a>, <a class="txt" href="#E28:21_2"><span class="lab"><font color="Green" title="E51">A43</font></span></a>, <a class="ref" href="card_2.html#T41" title="CARD_2:th.41">CARD_2:41</a></span>;<br/>
<a NAME="E33:21_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">B2</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="c6">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E30:21_2"><span class="lab"><font color="Green" title="E52">A44</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">A1</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><span class="kw"> such that </span><br/><a NAME="E35:21_2"/><span class="lab"><font color="Green" title="E53">A45</font></span>: 
<font color="Maroon" title="c9">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B2</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="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>
<a NAME="E36:21_2"/><span class="lab"><font color="Green" title="E54">A46</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">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="c7">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:21_2"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E37:21_2"/><span class="lab"><font color="Green" title="E55">A47</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:21_2"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E38:21_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:21_2"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E39:21_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:21_2"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="txt" href="#E36:21_2"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E40:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A48</font></span>: 
<font color="Maroon" title="c9">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E41:21_2"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">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="c9">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:21_2"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E42:21_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<a NAME="E43:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A49</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21_2"><span class="lab"><font color="Green" title="E45">A37</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">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><span class="kw"> such that </span><br/><a NAME="E45:21_2"/><span class="lab"><font color="Green" title="E58">A50</font></span>: 
<font color="Maroon" title="c10">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c9">A1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ;<br/>
<a NAME="E46:21_2"/><span class="lab"><font color="Green" title="E59">A51</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E37:21_2"><span class="lab"><font color="Green" title="E55">A47</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E47:21_2"/><span class="lab"><font color="Green" title="E60">A52</font></span>: 
<font color="Maroon" title="c10">U</font> is  <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">B1</font>, <font color="Maroon" title="c12">B2</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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">COMBGRAS:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U</font> &amp; <font color="Maroon" title="c12">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U</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="c11">B1</font>,<font color="Maroon" title="c12">B2</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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_2_2"/>
( <font color="Maroon" title="c11">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U</font> &amp; <font color="Maroon" title="c12">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c11">B1</font>,<font color="Maroon" title="c12">B2</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></span><br/>

<a NAME="E2:21_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c11">B1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">L</font> &amp; <font color="Maroon" title="c12">B2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:21_2"><span class="lab"><font color="Green" title="E59">A51</font></span></a>, <a class="txt" href="#E40:21_2"><span class="lab"><font color="Green" title="E56">A48</font></span></a>, <a class="txt" href="#E45:21_2"><span class="lab"><font color="Green" title="E58">A50</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E3:21_2_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">B1</font>,<font color="Maroon" title="c12">B2</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="c7">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:21_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="c11">B1</font>,<font color="Maroon" title="c12">B2</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E48:21_2"/>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:21_2"><span class="lab"><font color="Green" title="E58">A50</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E49:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A53</font></span>: 
<font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_2"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="txt" href="#E6:21_2"><span class="lab"><font color="Green" title="E41">A33</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<a NAME="E50:21_2"/>
<font color="Maroon" title="c9">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:21_2"><span class="lab"><font color="Green" title="E58">A50</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E51:21_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">U</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_2"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="txt" href="#E6:21_2"><span class="lab"><font color="Green" title="E41">A33</font></span></a>, <a class="txt" href="#E43:21_2"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E52:21_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E49:21_2"><span class="lab"><font color="Green" title="E61">A53</font></span></a>, <a class="txt" href="#E47:21_2"><span class="lab"><font color="Green" title="E60">A52</font></span></a>, <a class="ref" href="combgras.html#D3" target="_self" title="COMBGRAS:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E27:21"/>
 <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
 
;<br/>
<a NAME="E28:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A54</font></span>: 
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="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:21"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="txt" href="#E24:21"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">R</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E30:21"/><span class="lab"><font color="Green" title="E40">A55</font></span>: 
<font color="Maroon" title="c4">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">and </span><br/><a NAME="E31:21"/><span class="lab"><font color="Green" title="E41">A56</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></span>;<br/>
<a NAME="E32:21"/>
<font color="Maroon" title="c4">R</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:21"><span class="lab"><font color="Green" title="E41">A56</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">A1</font>, <font color="Maroon" title="c6">B1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E34:21"/><span class="lab"><font color="Green" title="E42">A57</font></span>: 
<font color="Maroon" title="c5">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">B1</font>
 <span class="kw">and </span><br/><a NAME="E35:21"/><span class="lab"><font color="Green" title="E43">A58</font></span>: 
<font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">A1</font>,<font color="Maroon" title="c6">B1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:21"><span class="lab"><font color="Green" title="E41">A56</font></span></a>, <a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></span>;<br/>
<a NAME="E36:21"/><span class="lab"><font color="Green" title="E44">A59</font></span>: 
<font color="Maroon" title="c5">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:21"><span class="lab"><font color="Green" title="E43">A58</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E37:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A60</font></span>: 
<font color="Maroon" title="c5">A1</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E40">A55</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">A</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><span class="kw"> such that </span><br/><a NAME="E39:21"/><span class="lab"><font color="Green" title="E46">A61</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font>
 ;<br/>
<a NAME="E40:21"/><span class="lab"><font color="Green" title="E47">A62</font></span>: 
<font color="Maroon" title="c6">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:21"><span class="lab"><font color="Green" title="E43">A58</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E41:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A63</font></span>: 
<font color="Maroon" title="c6">B1</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E40">A55</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">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><span class="kw"> such that </span><br/><a NAME="E43:21"/><span class="lab"><font color="Green" title="E49">A64</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B1</font>
 ;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E45:21"/><span class="lab"><font color="Green" title="E50">A65</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E40">A55</font></span></a>, <a class="txt" href="#E36:21"><span class="lab"><font color="Green" title="E44">A59</font></span></a>, <a class="txt" href="#E40:21"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E39:21"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E43:21"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></span>;<br/>
<a NAME="E46:21"/>
<font color="Maroon" title="c9">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/>
<a NAME="E47:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A66</font></span>: 
 ex <font color="Olive" title="b1">L1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">L</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">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 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E33">A14</font></span></a></span>;<br/>
<a NAME="E48:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A67</font></span>: 
<font color="Maroon" title="c9">L</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<a NAME="E49:21"/>
<font color="Maroon" title="c7">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:21"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E50:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A68</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E51:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A69</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T108" title="XBOOLE_1:th.108">XBOOLE_1:108</a></span>;<br/>
<a NAME="E52:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A70</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:21"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E53:21"/>
<font color="Maroon" title="c8">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:21"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E54:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A71</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E55:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A72</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:21"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E56:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A73</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:21"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E57:21"/><span class="lab"><font color="Green" title="E59">A74</font></span>: 
 ex <font color="Olive" title="b1">B2</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E41:21"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="txt" href="#E43:21"><span class="lab"><font color="Green" title="E49">A64</font></span></a></span>;<br/>
<a NAME="E58:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A75</font></span>: 
<font color="Maroon" title="c8">B</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<a NAME="E59:21"/><span class="lab"><font color="Green" title="E61">A76</font></span>: 
 ex <font color="Olive" title="b1">A2</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E37:21"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E39:21"><span class="lab"><font color="Green" title="E46">A61</font></span></a></span>;<br/>
<a NAME="E60:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A77</font></span>: 
<font color="Maroon" title="c7">A</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<a NAME="E61:21"/><span class="lab"><font color="Green" title="E63">A78</font></span>: 
<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>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:21"><span class="lab"><font color="Green" title="E42">A57</font></span></a>, <a class="txt" href="#E39:21"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E43:21"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E59:21"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E62:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A79</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:21"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E63:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A80</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:21"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E54:21"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E47:21"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E48:21"><span class="lab"><font color="Green" title="E52">A67</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E64:21"/><span class="lab"><font color="Green" title="E66">A81</font></span>: 
( (  for <font color="Olive" title="b1">C</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>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> or  not <font color="Olive" title="b1">C</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L</font> or  not <font color="Maroon" title="c7">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c8">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">C</font> ) ) implies <font color="Maroon" title="c3">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:21_3"/><span class="lab"><font color="Green" title="E67">A82</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c9">L</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="21_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_3_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c9">L</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_3_1"/><span class="kw">then </span>
<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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E47:21"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21_3_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:21_3"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c9">L</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> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E47:21"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E3:21_3"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c9">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_3"><span class="lab"><font color="Green" title="E67">A82</font></span></a>, <a class="ref" href="card_1.html#T3" title="CARD_1:th.3">CARD_1:3</a></span>;<br/>
<a NAME="E4:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">L</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T68" title="CARD_1:th.68">CARD_1:68</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:21_3"/><span class="lab"><font color="Green" title="E68">A83</font></span>: 
<font color="Maroon" title="c10">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="c9">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E7:21_3"/><span class="lab"><font color="Green" title="E69">A84</font></span>: 
(  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp;  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:21"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E54:21"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E6:21_3"><span class="lab"><font color="Green" title="E68">A83</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E8:21_3"/><span class="lab"><font color="Green" title="E70">A85</font></span>: 
( <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="21_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_3_2"/>
( <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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>  or <font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:21_3_2"/><span class="lab"><font color="Green" title="E71">A86</font></span>: 
( <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E4:21_3_2"/>
( ( <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c11">z1</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">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c11">z1</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">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_3_2"><span class="lab"><font color="Green" title="E71">A86</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:21_3_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_3"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:21_3"/><span class="lab"><font color="Green" title="E71">A87</font></span>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:21"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="txt" href="#E56:21"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E61:21"><span class="lab"><font color="Green" title="E63">A78</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E10:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A88</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E11:21_3"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span></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> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:21"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="txt" href="#E60:21"><span class="lab"><font color="Green" title="E62">A77</font></span></a>, <a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E13:21_3"/><span class="lab"><font color="Green" title="E73">A89</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">z1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T42" title="CARD_2:th.42">CARD_2:42</a></span>;<br/>
<a NAME="E14:21_3"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span></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> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E58:21"><span class="lab"><font color="Green" title="E60">A75</font></span></a>, <a class="txt" href="#E10:21_3"><span class="lab"><font color="Green" title="E72">A88</font></span></a>, <a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E16:21_3"/><span class="lab"><font color="Green" title="E74">A90</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">z2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T42" title="CARD_2:th.42">CARD_2:42</a></span>;<br/>
<a NAME="E17:21_3"/><span class="lab"><font color="Green" title="E75">A91</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">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="c12">z2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21_3"><span class="lab"><font color="Green" title="E74">A90</font></span></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#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></span>;<br/>
<a NAME="E18:21_3"/><span class="lab"><font color="Green" title="E76">A92</font></span>: 
<font color="Maroon" title="c12">z2</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="c12">z2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E19:21_3"/><span class="lab"><font color="Green" title="E77">A93</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:21"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E61:21"><span class="lab"><font color="Green" title="E63">A78</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E20:21_3"/><span class="lab"><font color="Green" title="E78">A94</font></span>: 
 not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:21"><span class="lab"><font color="Green" title="E54">A69</font></span></a>, <a class="txt" href="#E6:21_3"><span class="lab"><font color="Green" title="E68">A83</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E21:21_3"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:21"><span class="lab"><font color="Green" title="E61">A76</font></span></a></span>;<br/>
<a NAME="E22:21_3"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E19:21_3"><span class="lab"><font color="Green" title="E77">A93</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E23:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A95</font></span>: 
 <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="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">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="c10">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:21"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E48:21"><span class="lab"><font color="Green" title="E52">A67</font></span></a>, <a class="txt" href="#E20:21_3"><span class="lab"><font color="Green" title="E78">A94</font></span></a>, <a class="ref" href="card_2.html#T41" title="CARD_2:th.41">CARD_2:41</a></span>;<br/>
<a NAME="E24:21_3"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_3"><span class="lab"><font color="Green" title="E68">A83</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<a NAME="E25:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A96</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">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="c10">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:21"><span class="lab"><font color="Green" title="E55">A70</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E26:21_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">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="c10">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E23:21_3"><span class="lab"><font color="Green" title="E79">A95</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">C</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><span class="kw"> such that </span><br/><a NAME="E28:21_3"/><span class="lab"><font color="Green" title="E81">A97</font></span>: 
<font color="Maroon" title="c13">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">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="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>
<a NAME="E29:21_3"/><span class="lab"><font color="Green" title="E82">A98</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E25:21_3"><span class="lab"><font color="Green" title="E80">A96</font></span></a>, <a class="txt" href="#E28:21_3"><span class="lab"><font color="Green" title="E81">A97</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E30:21_3"/><span class="lab"><font color="Green" title="E83">A99</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:21"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="txt" href="#E25:21_3"><span class="lab"><font color="Green" title="E80">A96</font></span></a>, <a class="txt" href="#E28:21_3"><span class="lab"><font color="Green" title="E81">A97</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E31:21_3"/><span class="lab"><font color="Green" title="E84">A100</font></span>: 
1 <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> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E32:21_3"/><span class="lab"><font color="Green" title="E85">A101</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E33:21_3"/>
<font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c13">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">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> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">A</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:21_3"><span class="lab"><font color="Green" title="E81">A97</font></span></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>;<br/>
<a NAME="E34:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c13">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>;<br/>
<a NAME="E35:21_3"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c13">C</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E8:21_3"><span class="lab"><font color="Green" title="E70">A85</font></span></a>, <a class="txt" href="#E9:21_3"><span class="lab"><font color="Green" title="E71">A87</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E36:21_3"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">C</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E23:21_3"><span class="lab"><font color="Green" title="E79">A95</font></span></a>, <a class="txt" href="#E28:21_3"><span class="lab"><font color="Green" title="E81">A97</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E37:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">C</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="txt" href="#E29:21_3"><span class="lab"><font color="Green" title="E82">A98</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">L2</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><span class="kw"> such that </span><br/><a NAME="E39:21_3"/><span class="lab"><font color="Green" title="E86">A102</font></span>: 
<font color="Maroon" title="c14">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">C</font>
 ;<br/>
<a NAME="E40:21_3"/>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c13">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:21_3"><span class="lab"><font color="Green" title="E81">A97</font></span></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>;<br/>
<a NAME="E41:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c13">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>;<br/>
<a NAME="E42:21_3"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c13">C</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E8:21_3"><span class="lab"><font color="Green" title="E70">A85</font></span></a>, <a class="txt" href="#E9:21_3"><span class="lab"><font color="Green" title="E71">A87</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E43:21_3"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">C</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E59:21"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="txt" href="#E23:21_3"><span class="lab"><font color="Green" title="E79">A95</font></span></a>, <a class="txt" href="#E28:21_3"><span class="lab"><font color="Green" title="E81">A97</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E44:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">C</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="txt" href="#E30:21_3"><span class="lab"><font color="Green" title="E83">A99</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">L1</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><span class="kw"> such that </span><br/><a NAME="E46:21_3"/><span class="lab"><font color="Green" title="E87">A103</font></span>: 
<font color="Maroon" title="c15">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">C</font>
 ;<br/>
<a NAME="E47:21_3"/><span class="lab"><font color="Green" title="E88">A104</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> is  <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">Z1</font>, <font color="Maroon" title="c17">Z2</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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">COMBGRAS:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> &amp; <font color="Maroon" title="c17">Z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> 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="c16">Z1</font>,<font color="Maroon" title="c17">Z2</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> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21_3_3"/><span class="lab"><font color="Green" title="E89">A105</font></span>: 
<font color="Maroon" title="c16">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>
 <span class="kw">and </span><br/><a NAME="E2:21_3_3"/><span class="lab"><font color="Green" title="E90">A106</font></span>: 
<font color="Maroon" title="c17">Z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c16">Z1</font>,<font color="Maroon" title="c17">Z2</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></span><br/>

<a NAME="E3:21_3_3"/><span class="lab"><font color="Green" title="E91">A107</font></span>: 
( <font color="Maroon" title="c17">Z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> or <font color="Maroon" title="c17">Z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> or <font color="Maroon" title="c17">Z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_3_3"><span class="lab"><font color="Green" title="E90">A106</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/>
<a NAME="E4:21_3_3"/>
( <font color="Maroon" title="c16">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> or <font color="Maroon" title="c16">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> or <font color="Maroon" title="c16">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_3_3"><span class="lab"><font color="Green" title="E89">A105</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/>
<a NAME="E5:21_3_3"/><span class="kw">then </span>
( ( <font color="Maroon" title="c16">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c17">Z2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font> ) or ( <font color="Maroon" title="c16">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">C</font> &amp; <font color="Maroon" title="c17">Z2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">C</font> ) or ( <font color="Maroon" title="c16">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">C</font> &amp; <font color="Maroon" title="c17">Z2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">C</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_3_3"><span class="lab"><font color="Green" title="E91">A107</font></span></a>, <a class="ref" href="xboole_1.html#T11" title="XBOOLE_1:th.11">XBOOLE_1:11</a></span>;<br/>
<a NAME="E6:21_3_3"/><span class="kw">then </span>
( ( <font color="Maroon" title="c16">Z1</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="c17">Z2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L</font> ) or ( <font color="Maroon" title="c16">Z1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">L1</font> &amp; <font color="Maroon" title="c17">Z2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">L1</font> ) or ( <font color="Maroon" title="c16">Z1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">L2</font> &amp; <font color="Maroon" title="c17">Z2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">L2</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E63:21"><span class="lab"><font color="Green" title="E65">A80</font></span></a>, <a class="txt" href="#E46:21_3"><span class="lab"><font color="Green" title="E87">A103</font></span></a>, <a class="txt" href="#E39:21_3"><span class="lab"><font color="Green" title="E86">A102</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E7:21_3_3"/><span class="kw">then </span>
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c16">Z1</font>,<font color="Maroon" title="c17">Z2</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> or <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c16">Z1</font>,<font color="Maroon" title="c17">Z2</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="c15">L1</font> or <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c16">Z1</font>,<font color="Maroon" title="c17">Z2</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="c14">L2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:21_3_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="c16">Z1</font>,<font color="Maroon" title="c17">Z2</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E48:21_3"/><span class="lab"><font color="Green" title="E89">A108</font></span>: 
( <font color="Maroon" title="c13">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c13">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">B</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_3_4"/><span class="lab"><font color="Green" title="E90">A109</font></span>: 
( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">C</font> or <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_3_4"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">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="c13">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:21_3"><span class="lab"><font color="Green" title="E81">A97</font></span></a>, <a class="ref" href="xboole_1.html#T11" title="XBOOLE_1:th.11">XBOOLE_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21_3_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_3"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E1:21_3_4"><span class="lab"><font color="Green" title="E90">A109</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E49:21_3"/><span class="lab"><font color="Green" title="E90">A110</font></span>: 
3 <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="c3">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_3_5"/>
 not 3 <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="c3">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_3_5"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T16" title="ORDINAL1:th.16">ORDINAL1:16</a></span>;<br/>
<a NAME="E3:21_3_5"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">K</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>;<br/>
<a NAME="E4:21_3_5"/><span class="kw">then </span>
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">K</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:21"><span class="lab"><font color="Green" title="E39">A54</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E5:21_3_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A111</font></span>: 
<font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E40">A55</font></span></a>, <a class="txt" href="#E31:21"><span class="lab"><font color="Green" title="E41">A56</font></span></a>, <a class="txt" href="#E35:21"><span class="lab"><font color="Green" title="E43">A58</font></span></a>, <a class="txt" href="#E39:21"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E43:21"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a></span>;<br/>
<a NAME="E6:21_3_5"/>
( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> &amp; <font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/>
<a NAME="E7:21_3_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A112</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/>
<a NAME="E8:21_3_5"/>
<font color="Maroon" title="c13">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/>
<a NAME="E9:21_3_5"/><span class="kw">then </span>
 not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:21_3"><span class="lab"><font color="Green" title="E89">A108</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:21_3_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E47:21_3"><span class="lab"><font color="Green" title="E88">A104</font></span></a>, <a class="txt" href="#E5:21_3_5"><span class="lab"><font color="Green" title="E91">A111</font></span></a>, <a class="txt" href="#E7:21_3_5"><span class="lab"><font color="Green" title="E92">A112</font></span></a>, <a class="ref" href="combgras.html#D3" target="_self" title="COMBGRAS:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E50:21_3"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <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="c3">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_3_6"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <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="c3">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_3_6"/><span class="kw">then </span>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E31:21"><span class="lab"><font color="Green" title="E41">A56</font></span></a>, <a class="txt" href="#E35:21"><span class="lab"><font color="Green" title="E43">A58</font></span></a>, <a class="txt" href="#E39:21"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E43:21"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E49:21_3"><span class="lab"><font color="Green" title="E90">A110</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21_3_6"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E51:21_3"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:21"><span class="lab"><font color="Green" title="E39">A54</font></span></a>, <a class="txt" href="#E31:21"><span class="lab"><font color="Green" title="E41">A56</font></span></a>, <a class="txt" href="#E35:21"><span class="lab"><font color="Green" title="E43">A58</font></span></a>, <a class="txt" href="#E39:21"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E43:21"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="ref" href="card_1.html#T3" title="CARD_1:th.3">CARD_1:3</a></span>;<br/>
<a NAME="E52:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">K</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T68" title="CARD_1:th.68">CARD_1:68</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">E2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E54:21_3"/><span class="lab"><font color="Green" title="E91">A113</font></span>: 
<font color="Maroon" title="c16">E2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E55:21_3"/><span class="lab"><font color="Green" title="E92">A114</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:21"><span class="lab"><font color="Green" title="E61">A76</font></span></a></span>;<br/>
<a NAME="E56:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A115</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">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="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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E19:21_3"><span class="lab"><font color="Green" title="E77">A93</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E57:21_3"/><span class="lab"><font color="Green" title="E94">A116</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c8">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a></span>;<br/>
<a NAME="E58:21_3"/><span class="lab"><font color="Green" title="E95">A117</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E59:21_3"/><span class="lab"><font color="Green" title="E96">A118</font></span>: 
 not <font color="Maroon" title="c16">E2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:21_3"><span class="lab"><font color="Green" title="E91">A113</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E60:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A119</font></span>: 
<font color="Maroon" title="c16">E2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E61:21_3"/>
<font color="Maroon" title="c16">E2</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:21_3"><span class="lab"><font color="Green" title="E91">A113</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">E1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><span class="kw"> such that </span><br/><a NAME="E63:21_3"/><span class="lab"><font color="Green" title="E98">A120</font></span>: 
<font color="Maroon" title="c17">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">E2</font>
 <span class="kw">and </span><br/><a NAME="E64:21_3"/><span class="lab"><font color="Green" title="E99">A121</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c17">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E26">A7</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">E</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><span class="kw"> such that </span><br/><a NAME="E66:21_3"/><span class="lab"><font color="Green" title="E100">A122</font></span>: 
<font color="Maroon" title="c18">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">E1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:21_3"><span class="lab"><font color="Green" title="E91">A113</font></span></a>, <a class="txt" href="#E63:21_3"><span class="lab"><font color="Green" title="E98">A120</font></span></a></span>;<br/>
<a NAME="E67:21_3"/><span class="lab"><font color="Green" title="E101">A123</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">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="c11">z1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_3"><span class="lab"><font color="Green" title="E73">A89</font></span></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#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></span>;<br/>
<a NAME="E68:21_3"/><span class="lab"><font color="Green" title="E102">A124</font></span>: 
<font color="Maroon" title="c11">z1</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="c11">z1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E69:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A125</font></span>: 
 not <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_3"><span class="lab"><font color="Green" title="E73">A89</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E70:21_3"/><span class="lab"><font color="Green" title="E104">A126</font></span>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">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="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; 2 <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> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E59:21"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">S</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E72:21_3"/><span class="lab"><font color="Green" title="E105">A127</font></span>: 
<font color="Maroon" title="c19">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">D</font> where <font color="Olive" title="b1">D</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">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">D</font> ) </span> } </span> 
 ;<br/>
<a NAME="E73:21_3"/><span class="lab"><font color="Green" title="E106">A128</font></span>: 
<font color="Maroon" title="c16">E2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:21_3"><span class="lab"><font color="Green" title="E91">A113</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">K1</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><span class="kw"> such that </span><br/><a NAME="E75:21_3"/><span class="lab"><font color="Green" title="E107">A129</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c18">E</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="c20">K1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E40">A55</font></span></a>, <a class="txt" href="#E36:21"><span class="lab"><font color="Green" title="E44">A59</font></span></a>, <a class="txt" href="#E39:21"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E63:21_3"><span class="lab"><font color="Green" title="E98">A120</font></span></a>, <a class="txt" href="#E66:21_3"><span class="lab"><font color="Green" title="E100">A122</font></span></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c21">K2</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><span class="kw"> such that </span><br/><a NAME="E77:21_3"/><span class="lab"><font color="Green" title="E108">A130</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c18">E</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="c21">K2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E40">A55</font></span></a>, <a class="txt" href="#E40:21"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E43:21"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E73:21_3"><span class="lab"><font color="Green" title="E106">A128</font></span></a>, <a class="txt" href="#E63:21_3"><span class="lab"><font color="Green" title="E98">A120</font></span></a>, <a class="txt" href="#E66:21_3"><span class="lab"><font color="Green" title="E100">A122</font></span></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></span>;<br/>
<a NAME="E78:21_3"/>
<font color="Maroon" title="c18">E</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">K2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E77:21_3"><span class="lab"><font color="Green" title="E108">A130</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E79:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A131</font></span>: 
<font color="Maroon" title="c18">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">K2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E80:21_3"/>
<font color="Maroon" title="c21">K2</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/>
<a NAME="E81:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A132</font></span>: 
 ex <font color="Olive" title="b1">M2</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Maroon" title="c21">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M2</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">M2</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E33">A14</font></span></a></span>;<br/>
<a NAME="E82:21_3"/>
<font color="Maroon" title="c8">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">K2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E77:21_3"><span class="lab"><font color="Green" title="E108">A130</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E83:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">K2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E84:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">K2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:21_3"><span class="lab"><font color="Green" title="E109">A131</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E85:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A133</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">E</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:21_3"><span class="lab"><font color="Green" title="E110">A132</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E86:21_3"/><span class="lab"><font color="Green" title="E112">A134</font></span>: 
<font color="Maroon" title="c16">E2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:21_3"><span class="lab"><font color="Green" title="E96">A118</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E87:21_3"/><span class="kw">then </span>
<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>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">E</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E63:21_3"><span class="lab"><font color="Green" title="E98">A120</font></span></a>, <a class="txt" href="#E64:21_3"><span class="lab"><font color="Green" title="E99">A121</font></span></a>, <a class="txt" href="#E66:21_3"><span class="lab"><font color="Green" title="E100">A122</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E88:21_3"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">E</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:21_3"><span class="lab"><font color="Green" title="E111">A133</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E89:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A135</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">E</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E64:21_3"><span class="lab"><font color="Green" title="E99">A121</font></span></a>, <a class="txt" href="#E66:21_3"><span class="lab"><font color="Green" title="E100">A122</font></span></a>, <a class="txt" href="#E57:21_3"><span class="lab"><font color="Green" title="E94">A116</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E90:21_3"/>
 for <font color="Olive" title="b1">C</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>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> or  not <font color="Olive" title="b1">C</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L</font> or  not <font color="Maroon" title="c7">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c8">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> </span><br/>

<a NAME="E91:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A136</font></span>: 
 not <font color="Maroon" title="c18">E</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E73:21_3"><span class="lab"><font color="Green" title="E106">A128</font></span></a>, <a class="txt" href="#E60:21_3"><span class="lab"><font color="Green" title="E97">A119</font></span></a>, <a class="txt" href="#E86:21_3"><span class="lab"><font color="Green" title="E112">A134</font></span></a>, <a class="txt" href="#E63:21_3"><span class="lab"><font color="Green" title="E98">A120</font></span></a>, <a class="txt" href="#E66:21_3"><span class="lab"><font color="Green" title="E100">A122</font></span></a></span>;<br/>
<a NAME="E92:21_3"/><span class="lab"><font color="Green" title="E115">A137</font></span>: 
 not  <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="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">E</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
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_3_7"/><span class="lab"><font color="Green" title="E116">A138</font></span>: 
 <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="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">E</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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_3_7"/><span class="kw">then </span>
( <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">E</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">E</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E3:21_3_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A139</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:21"><span class="lab"><font color="Green" title="E64">A79</font></span></a>, <a class="txt" href="#E1:21_3_7"><span class="lab"><font color="Green" title="E116">A138</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a></span>;<br/>
<a NAME="E4:21_3_7"/>
<font color="Maroon" title="c18">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E5:21_3_7"/><span class="kw">then </span>
<font color="Maroon" title="c18">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:21"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E3:21_3_7"><span class="lab"><font color="Green" title="E117">A139</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:21_3_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E91:21_3"><span class="lab"><font color="Green" title="E114">A136</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E93:21_3"/>
<font color="Maroon" title="c18">E</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">K1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:21_3"><span class="lab"><font color="Green" title="E107">A129</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E94:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A140</font></span>: 
<font color="Maroon" title="c18">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">K1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E95:21_3"/>
<font color="Maroon" title="c20">K1</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/>
<a NAME="E96:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A141</font></span>: 
 ex <font color="Olive" title="b1">M1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Maroon" title="c20">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">M1</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E33">A14</font></span></a></span>;<br/>
<a NAME="E97:21_3"/>
<font color="Maroon" title="c7">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">K1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:21_3"><span class="lab"><font color="Green" title="E107">A129</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E98:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">K1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E99:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">K1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E94:21_3"><span class="lab"><font color="Green" title="E116">A140</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E100:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A142</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">E</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E96:21_3"><span class="lab"><font color="Green" title="E117">A141</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E101:21_3"/>
<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>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">E</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:21"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="txt" href="#E60:21_3"><span class="lab"><font color="Green" title="E97">A119</font></span></a>, <a class="txt" href="#E63:21_3"><span class="lab"><font color="Green" title="E98">A120</font></span></a>, <a class="txt" href="#E64:21_3"><span class="lab"><font color="Green" title="E99">A121</font></span></a>, <a class="txt" href="#E66:21_3"><span class="lab"><font color="Green" title="E100">A122</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E102:21_3"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">E</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E100:21_3"><span class="lab"><font color="Green" title="E118">A142</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E103:21_3"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">E</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E64:21_3"><span class="lab"><font color="Green" title="E99">A121</font></span></a>, <a class="txt" href="#E66:21_3"><span class="lab"><font color="Green" title="E100">A122</font></span></a>, <a class="txt" href="#E55:21_3"><span class="lab"><font color="Green" title="E92">A114</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E104:21_3"/><span class="kw">then </span>
( (  <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="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">E</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 &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="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">E</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="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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">E</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> 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="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">E</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 ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E64:21_3"><span class="lab"><font color="Green" title="E99">A121</font></span></a>, <a class="txt" href="#E66:21_3"><span class="lab"><font color="Green" title="E100">A122</font></span></a>, <a class="txt" href="#E89:21_3"><span class="lab"><font color="Green" title="E113">A135</font></span></a>, <a class="txt" href="#E70:21_3"><span class="lab"><font color="Green" title="E104">A126</font></span></a>, <a class="txt" href="#E31:21_3"><span class="lab"><font color="Green" title="E84">A100</font></span></a>, <a class="txt" href="#E56:21_3"><span class="lab"><font color="Green" title="E93">A115</font></span></a>, <a class="ref" href="combgras.html#T7" target="_self" title="COMBGRAS:th.7">Th7</a></span>;<br/>
<a NAME="E105:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A143</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:21"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E48:21"><span class="lab"><font color="Green" title="E52">A67</font></span></a>, <a class="txt" href="#E10:21_3"><span class="lab"><font color="Green" title="E72">A88</font></span></a>, <a class="txt" href="#E92:21_3"><span class="lab"><font color="Green" title="E115">A137</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E106:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A144</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E107:21_3"/>
<font color="Maroon" title="c18">E</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:21_3"><span class="lab"><font color="Green" title="E99">A121</font></span></a>, <a class="txt" href="#E66:21_3"><span class="lab"><font color="Green" title="E100">A122</font></span></a></span>;<br/>
<a NAME="E108:21_3"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span></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> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_3"><span class="lab"><font color="Green" title="E72">A88</font></span></a>, <a class="txt" href="#E64:21_3"><span class="lab"><font color="Green" title="E99">A121</font></span></a>, <a class="txt" href="#E66:21_3"><span class="lab"><font color="Green" title="E100">A122</font></span></a>, <a class="txt" href="#E105:21_3"><span class="lab"><font color="Green" title="E119">A143</font></span></a>, <a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">z4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E110:21_3"/><span class="lab"><font color="Green" title="E121">A145</font></span>: 
<font color="Maroon" title="c18">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c22">z4</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T42" title="CARD_2:th.42">CARD_2:42</a></span>;<br/>
<a NAME="E111:21_3"/><span class="lab"><font color="Green" title="E122">A146</font></span>: 
<font color="Maroon" title="c18">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">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="c22">z4</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E105:21_3"><span class="lab"><font color="Green" title="E119">A143</font></span></a>, <a class="txt" href="#E110:21_3"><span class="lab"><font color="Green" title="E121">A145</font></span></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#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></span>;<br/>
<a NAME="E112:21_3"/><span class="lab"><font color="Green" title="E123">A147</font></span>: 
<font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_3_8"/>
 not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">D2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:21_3_8"/><span class="lab"><font color="Green" title="E124">A148</font></span>: 
<font color="Maroon" title="c23">D2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">and </span><br/><a NAME="E4:21_3_8"/><span class="lab"><font color="Green" title="E125">A149</font></span>: 
 not <font color="Maroon" title="c23">D2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E5:21_3_8"/>
<font color="Maroon" title="c23">D2</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_3_8"><span class="lab"><font color="Green" title="E124">A148</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">D1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><span class="kw"> such that </span><br/><a NAME="E7:21_3_8"/><span class="lab"><font color="Green" title="E126">A150</font></span>: 
<font color="Maroon" title="c24">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">D2</font>
 <span class="kw">and </span><br/><a NAME="E8:21_3_8"/><span class="lab"><font color="Green" title="E127">A151</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c24">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E26">A7</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">D</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><span class="kw"> such that </span><br/><a NAME="E10:21_3_8"/><span class="lab"><font color="Green" title="E128">A152</font></span>: 
<font color="Maroon" title="c25">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_3_8"><span class="lab"><font color="Green" title="E124">A148</font></span></a>, <a class="txt" href="#E7:21_3_8"><span class="lab"><font color="Green" title="E126">A150</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c26">K11</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><span class="kw"> such that </span><br/><a NAME="E12:21_3_8"/><span class="lab"><font color="Green" title="E129">A153</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c25">D</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="c26">K11</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E40">A55</font></span></a>, <a class="txt" href="#E36:21"><span class="lab"><font color="Green" title="E44">A59</font></span></a>, <a class="txt" href="#E39:21"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E3:21_3_8"><span class="lab"><font color="Green" title="E124">A148</font></span></a>, <a class="txt" href="#E7:21_3_8"><span class="lab"><font color="Green" title="E126">A150</font></span></a>, <a class="txt" href="#E10:21_3_8"><span class="lab"><font color="Green" title="E128">A152</font></span></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></span>;<br/>
<a NAME="E13:21_3_8"/>
<font color="Maroon" title="c25">D</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c26">K11</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21_3_8"><span class="lab"><font color="Green" title="E129">A153</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E14:21_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A154</font></span>: 
<font color="Maroon" title="c25">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c26">K11</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E15:21_3_8"/>
<font color="Maroon" title="c26">K11</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/>
<a NAME="E16:21_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A155</font></span>: 
 ex <font color="Olive" title="b1">R11</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">R11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">K11</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">R11</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E33">A14</font></span></a></span>;<br/>
<a NAME="E17:21_3_8"/><span class="lab"><font color="Green" title="E132">A156</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c25">D</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_3_8"><span class="lab"><font color="Green" title="E127">A151</font></span></a>, <a class="txt" href="#E10:21_3_8"><span class="lab"><font color="Green" title="E128">A152</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">K13</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><span class="kw"> such that </span><br/><a NAME="E19:21_3_8"/><span class="lab"><font color="Green" title="E133">A157</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">E</font>,<font color="Maroon" title="c25">D</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="c27">K13</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E73:21_3"><span class="lab"><font color="Green" title="E106">A128</font></span></a>, <a class="txt" href="#E63:21_3"><span class="lab"><font color="Green" title="E98">A120</font></span></a>, <a class="txt" href="#E66:21_3"><span class="lab"><font color="Green" title="E100">A122</font></span></a>, <a class="txt" href="#E3:21_3_8"><span class="lab"><font color="Green" title="E124">A148</font></span></a>, <a class="txt" href="#E7:21_3_8"><span class="lab"><font color="Green" title="E126">A150</font></span></a>, <a class="txt" href="#E10:21_3_8"><span class="lab"><font color="Green" title="E128">A152</font></span></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c28">K12</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><span class="kw"> such that </span><br/><a NAME="E21:21_3_8"/><span class="lab"><font color="Green" title="E134">A158</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c25">D</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="c28">K12</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E40">A55</font></span></a>, <a class="txt" href="#E40:21"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E43:21"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E3:21_3_8"><span class="lab"><font color="Green" title="E124">A148</font></span></a>, <a class="txt" href="#E7:21_3_8"><span class="lab"><font color="Green" title="E126">A150</font></span></a>, <a class="txt" href="#E10:21_3_8"><span class="lab"><font color="Green" title="E128">A152</font></span></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></span>;<br/>
<a NAME="E22:21_3_8"/>
<font color="Maroon" title="c7">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c26">K11</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21_3_8"><span class="lab"><font color="Green" title="E129">A153</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E23:21_3_8"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c26">K11</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E24:21_3_8"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c25">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c26">K11</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21_3_8"><span class="lab"><font color="Green" title="E130">A154</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E25:21_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A159</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c25">D</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21_3_8"><span class="lab"><font color="Green" title="E131">A155</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E26:21_3_8"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E72:21_3"><span class="lab"><font color="Green" title="E105">A127</font></span></a>, <a class="txt" href="#E58:21_3"><span class="lab"><font color="Green" title="E95">A117</font></span></a>, <a class="txt" href="#E4:21_3_8"><span class="lab"><font color="Green" title="E125">A149</font></span></a>, <a class="txt" href="#E7:21_3_8"><span class="lab"><font color="Green" title="E126">A150</font></span></a>, <a class="txt" href="#E8:21_3_8"><span class="lab"><font color="Green" title="E127">A151</font></span></a>, <a class="txt" href="#E10:21_3_8"><span class="lab"><font color="Green" title="E128">A152</font></span></a></span>;<br/>
<a NAME="E27:21_3_8"/><span class="kw">then </span>
<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>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c25">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:21"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="txt" href="#E8:21_3_8"><span class="lab"><font color="Green" title="E127">A151</font></span></a>, <a class="txt" href="#E10:21_3_8"><span class="lab"><font color="Green" title="E128">A152</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E28:21_3_8"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c25">D</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21_3_8"><span class="lab"><font color="Green" title="E135">A159</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E29:21_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">A160</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E59:21"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="txt" href="#E17:21_3_8"><span class="lab"><font color="Green" title="E132">A156</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E30:21_3_8"/>
 not <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E72:21_3"><span class="lab"><font color="Green" title="E105">A127</font></span></a>, <a class="txt" href="#E4:21_3_8"><span class="lab"><font color="Green" title="E125">A149</font></span></a>, <a class="txt" href="#E7:21_3_8"><span class="lab"><font color="Green" title="E126">A150</font></span></a>, <a class="txt" href="#E8:21_3_8"><span class="lab"><font color="Green" title="E127">A151</font></span></a>, <a class="txt" href="#E10:21_3_8"><span class="lab"><font color="Green" title="E128">A152</font></span></a></span>;<br/>
<a NAME="E31:21_3_8"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font> &amp;  not <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E32:21_3_8"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E33:21_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A161</font></span>: 
 <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="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:21"><span class="lab"><font color="Green" title="E62">A77</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E34:21_3_8"/>
<font color="Maroon" title="c25">D</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c27">K13</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:21_3_8"><span class="lab"><font color="Green" title="E133">A157</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E35:21_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A162</font></span>: 
<font color="Maroon" title="c25">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">K13</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E36:21_3_8"/>
<font color="Maroon" title="c27">K13</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/>
<a NAME="E37:21_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E139">A163</font></span>: 
 ex <font color="Olive" title="b1">R13</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">R13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">K13</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">R13</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E33">A14</font></span></a></span>;<br/>
<a NAME="E38:21_3_8"/>
<font color="Maroon" title="c25">D</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c28">K12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:21_3_8"><span class="lab"><font color="Green" title="E134">A158</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E39:21_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A164</font></span>: 
<font color="Maroon" title="c25">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c28">K12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E40:21_3_8"/>
<font color="Maroon" title="c28">K12</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/>
<a NAME="E41:21_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A165</font></span>: 
 ex <font color="Olive" title="b1">R12</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">R12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">K12</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">R12</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E33">A14</font></span></a></span>;<br/>
<a NAME="E42:21_3_8"/>
<font color="Maroon" title="c8">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c28">K12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:21_3_8"><span class="lab"><font color="Green" title="E134">A158</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E43:21_3_8"/><span class="kw">then </span>
<font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c28">K12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E44:21_3_8"/><span class="kw">then </span>
<font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c25">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c28">K12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:21_3_8"><span class="lab"><font color="Green" title="E140">A164</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E45:21_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">A166</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c25">D</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:21_3_8"><span class="lab"><font color="Green" title="E141">A165</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E46:21_3_8"/>
<font color="Maroon" title="c8">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E72:21_3"><span class="lab"><font color="Green" title="E105">A127</font></span></a>, <a class="txt" href="#E32:21_3"><span class="lab"><font color="Green" title="E85">A101</font></span></a>, <a class="txt" href="#E4:21_3_8"><span class="lab"><font color="Green" title="E125">A149</font></span></a>, <a class="txt" href="#E7:21_3_8"><span class="lab"><font color="Green" title="E126">A150</font></span></a>, <a class="txt" href="#E8:21_3_8"><span class="lab"><font color="Green" title="E127">A151</font></span></a>, <a class="txt" href="#E10:21_3_8"><span class="lab"><font color="Green" title="E128">A152</font></span></a></span>;<br/>
<a NAME="E47:21_3_8"/><span class="kw">then </span>
<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>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c25">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E8:21_3_8"><span class="lab"><font color="Green" title="E127">A151</font></span></a>, <a class="txt" href="#E10:21_3_8"><span class="lab"><font color="Green" title="E128">A152</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E48:21_3_8"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c25">D</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:21_3_8"><span class="lab"><font color="Green" title="E142">A166</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E49:21_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A167</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E17:21_3_8"><span class="lab"><font color="Green" title="E132">A156</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E50:21_3_8"/>
<font color="Maroon" title="c18">E</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c27">K13</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:21_3_8"><span class="lab"><font color="Green" title="E133">A157</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E51:21_3_8"/><span class="kw">then </span>
<font color="Maroon" title="c18">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">K13</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E52:21_3_8"/><span class="kw">then </span>
<font color="Maroon" title="c18">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c25">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">K13</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:21_3_8"><span class="lab"><font color="Green" title="E138">A162</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E53:21_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E144">A168</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c25">D</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:21_3_8"><span class="lab"><font color="Green" title="E139">A163</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E54:21_3_8"/>
<font color="Maroon" title="c18">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E72:21_3"><span class="lab"><font color="Green" title="E105">A127</font></span></a>, <a class="txt" href="#E106:21_3"><span class="lab"><font color="Green" title="E120">A144</font></span></a>, <a class="txt" href="#E4:21_3_8"><span class="lab"><font color="Green" title="E125">A149</font></span></a>, <a class="txt" href="#E7:21_3_8"><span class="lab"><font color="Green" title="E126">A150</font></span></a>, <a class="txt" href="#E8:21_3_8"><span class="lab"><font color="Green" title="E127">A151</font></span></a>, <a class="txt" href="#E10:21_3_8"><span class="lab"><font color="Green" title="E128">A152</font></span></a></span>;<br/>
<a NAME="E55:21_3_8"/><span class="kw">then </span>
<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>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c25">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:21_3"><span class="lab"><font color="Green" title="E99">A121</font></span></a>, <a class="txt" href="#E66:21_3"><span class="lab"><font color="Green" title="E100">A122</font></span></a>, <a class="txt" href="#E8:21_3_8"><span class="lab"><font color="Green" title="E127">A151</font></span></a>, <a class="txt" href="#E10:21_3_8"><span class="lab"><font color="Green" title="E128">A152</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E56:21_3_8"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c25">D</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:21_3_8"><span class="lab"><font color="Green" title="E144">A168</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E57:21_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E145">A169</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E64:21_3"><span class="lab"><font color="Green" title="E99">A121</font></span></a>, <a class="txt" href="#E66:21_3"><span class="lab"><font color="Green" title="E100">A122</font></span></a>, <a class="txt" href="#E17:21_3_8"><span class="lab"><font color="Green" title="E132">A156</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E58:21_3_8"/><span class="lab"><font color="Green" title="E146">A170</font></span>: 
( <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">D</font> &amp; <font color="Maroon" title="c12">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">D</font> &amp; <font color="Maroon" title="c22">z4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">D</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_3_8_1"/>
(  not <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">D</font> or  not <font color="Maroon" title="c12">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">D</font> or  not <font color="Maroon" title="c22">z4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">D</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_3_8_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">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="c11">z1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">z1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c25">D</font> ) or ( <font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">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="c12">z2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">z2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c25">D</font> ) or ( <font color="Maroon" title="c18">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">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="c22">z4</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c22">z4</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c25">D</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_3"><span class="lab"><font color="Green" title="E73">A89</font></span></a>, <a class="txt" href="#E16:21_3"><span class="lab"><font color="Green" title="E74">A90</font></span></a>, <a class="txt" href="#E105:21_3"><span class="lab"><font color="Green" title="E119">A143</font></span></a>, <a class="txt" href="#E110:21_3"><span class="lab"><font color="Green" title="E121">A145</font></span></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#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a>, <a class="ref" href="zfmisc_1.html#T50" title="ZFMISC_1:th.50">ZFMISC_1:50</a></span>;<br/>
<a NAME="E3:21_3_8_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</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="c11">z1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">z1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) or ( <font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</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="c12">z2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">z2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) or ( <font color="Maroon" title="c18">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</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="c22">z4</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c22">z4</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 
<span class="kw">by</span> <span class="lab"><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#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:21_3_8_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E9:21_3"><span class="lab"><font color="Green" title="E71">A87</font></span></a>, <a class="txt" href="#E29:21_3_8"><span class="lab"><font color="Green" title="E136">A160</font></span></a>, <a class="txt" href="#E49:21_3_8"><span class="lab"><font color="Green" title="E143">A167</font></span></a>, <a class="txt" href="#E57:21_3_8"><span class="lab"><font color="Green" title="E145">A169</font></span></a>, <a class="txt" href="#E33:21_3_8"><span class="lab"><font color="Green" title="E137">A161</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E59:21_3_8"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c11">z1</font>,<font color="Maroon" title="c12">z2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">D</font> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c22">z4</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="c25">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/>
<a NAME="E60:21_3_8"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c11">z1</font>,<font color="Maroon" title="c12">z2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></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="c22">z4</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="c25">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E61:21_3_8"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">D</font> &amp; <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c11">z1</font>,<font color="Maroon" title="c12">z2</font>,<font color="Maroon" title="c22">z4</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T3" title="ENUMSET1:th.3">ENUMSET1:3</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E62:21_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E147">A171</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c11">z1</font>,<font color="Maroon" title="c12">z2</font>,<font color="Maroon" title="c22">z4</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E63:21_3_8"/><span class="lab"><font color="Green" title="E148">A172</font></span>: 
( <font color="Maroon" title="c22">z4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E110:21_3"><span class="lab"><font color="Green" title="E121">A145</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E64:21_3_8"/><span class="lab"><font color="Green" title="E149">A173</font></span>: 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c11">z1</font>,<font color="Maroon" title="c12">z2</font>,<font color="Maroon" title="c22">z4</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_8_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_3_8_2"/>
 not <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c11">z1</font>,<font color="Maroon" title="c12">z2</font>,<font color="Maroon" title="c22">z4</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_3_8_2"/><span class="kw">then </span>
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c11">z1</font>,<font color="Maroon" title="c12">z2</font>,<font color="Maroon" title="c22">z4</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:21_3_8_2"/><span class="lab"><font color="Green" title="E150">A174</font></span>: 
<font color="Maroon" title="c29">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c11">z1</font>,<font color="Maroon" title="c12">z2</font>,<font color="Maroon" title="c22">z4</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E5:21_3_8_2"/>
<font color="Maroon" title="c29">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c11">z1</font>,<font color="Maroon" title="c12">z2</font>,<font color="Maroon" title="c22">z4</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_3_8_2"><span class="lab"><font color="Green" title="E150">A174</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E6:21_3_8_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E151">A175</font></span>: 
( <font color="Maroon" title="c29">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z1</font> or <font color="Maroon" title="c29">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font> or <font color="Maroon" title="c29">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">z4</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/>
<a NAME="E7:21_3_8_2"/>
<font color="Maroon" title="c29">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_3_8_2"><span class="lab"><font color="Green" title="E150">A174</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:21_3_8_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_3"><span class="lab"><font color="Green" title="E73">A89</font></span></a>, <a class="txt" href="#E16:21_3"><span class="lab"><font color="Green" title="E74">A90</font></span></a>, <a class="txt" href="#E68:21_3"><span class="lab"><font color="Green" title="E102">A124</font></span></a>, <a class="txt" href="#E18:21_3"><span class="lab"><font color="Green" title="E76">A92</font></span></a>, <a class="txt" href="#E63:21_3_8"><span class="lab"><font color="Green" title="E148">A172</font></span></a>, <a class="txt" href="#E6:21_3_8_2"><span class="lab"><font color="Green" title="E151">A175</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c29">r</font> =  <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="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font></span>)</span> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:21"><span class="lab"><font color="Green" title="E62">A77</font></span></a></span>;<br/>
<a NAME="E66:21_3_8"/><span class="lab"><font color="Green" title="E150">A176</font></span>: 
 not <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:21_3"><span class="lab"><font color="Green" title="E103">A125</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E67:21_3_8"/>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">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="c11">z1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_3"><span class="lab"><font color="Green" title="E73">A89</font></span></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#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></span>;<br/>
<a NAME="E68:21_3_8"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</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="c11">z1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>;<br/>
<a NAME="E69:21_3_8"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</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="c11">z1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:21_3_8"><span class="lab"><font color="Green" title="E146">A170</font></span></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></span>;<br/>
<a NAME="E70:21_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E151">A177</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">r</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:21"><span class="lab"><font color="Green" title="E62">A77</font></span></a>, <a class="txt" href="#E66:21_3_8"><span class="lab"><font color="Green" title="E150">A176</font></span></a>, <a class="ref" href="card_2.html#T41" title="CARD_2:th.41">CARD_2:41</a></span>;<br/>
<a NAME="E71:21_3_8"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c11">z1</font>,<font color="Maroon" title="c12">z2</font>,<font color="Maroon" title="c22">z4</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:21"><span class="lab"><font color="Green" title="E42">A57</font></span></a>, <a class="txt" href="#E39:21"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E43:21"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E67:21_3"><span class="lab"><font color="Green" title="E101">A123</font></span></a>, <a class="txt" href="#E17:21_3"><span class="lab"><font color="Green" title="E75">A91</font></span></a>, <a class="txt" href="#E60:21_3"><span class="lab"><font color="Green" title="E97">A119</font></span></a>, <a class="txt" href="#E86:21_3"><span class="lab"><font color="Green" title="E112">A134</font></span></a>, <a class="txt" href="#E63:21_3"><span class="lab"><font color="Green" title="E98">A120</font></span></a>, <a class="txt" href="#E66:21_3"><span class="lab"><font color="Green" title="E100">A122</font></span></a>, <a class="txt" href="#E111:21_3"><span class="lab"><font color="Green" title="E122">A146</font></span></a>, <a class="ref" href="card_2.html#T58" title="CARD_2:th.58">CARD_2:58</a></span>;<br/>
<a NAME="E72:21_3_8"/><span class="kw">then </span>
 <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"><span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">D</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c11">z1</font>,<font color="Maroon" title="c12">z2</font>,<font color="Maroon" title="c22">z4</font></span><a href="enumset1.html#K1" title="ENUMSET1: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> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:21"><span class="lab"><font color="Green" title="E62">A77</font></span></a>, <a class="txt" href="#E29:21_3_8"><span class="lab"><font color="Green" title="E136">A160</font></span></a>, <a class="txt" href="#E70:21_3_8"><span class="lab"><font color="Green" title="E151">A177</font></span></a>, <a class="txt" href="#E64:21_3_8"><span class="lab"><font color="Green" title="E149">A173</font></span></a>, <a class="ref" href="card_2.html#T40" title="CARD_2:th.40">CARD_2:40</a></span>;<br/>
<a NAME="E73:21_3_8"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_3_8"><span class="lab"><font color="Green" title="E127">A151</font></span></a>, <a class="txt" href="#E10:21_3_8"><span class="lab"><font color="Green" title="E128">A152</font></span></a>, <a class="txt" href="#E62:21_3_8"><span class="lab"><font color="Green" title="E147">A171</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E74:21_3_8"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E75:21_3_8"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E113:21_3"/>
<font color="Maroon" title="c19">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c23">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">S</font> or <font color="Maroon" title="c23">Z</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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_3_9"/>
<font color="Maroon" title="c23">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">Z</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></span><br/>

<a NAME="E2:21_3_9"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">Z1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Maroon" title="c23">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Z1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E72:21_3"><span class="lab"><font color="Green" title="E105">A127</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21_3_9"/>
<font color="Maroon" title="c23">Z</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E26">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">S1</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><span class="kw"> such that </span><br/><a NAME="E115:21_3"/><span class="lab"><font color="Green" title="E124">A178</font></span>: 
<font color="Maroon" title="c23">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">S</font>
 ;<br/>
<a NAME="E116:21_3"/><span class="lab"><font color="Green" title="E125">A179</font></span>: 
<font color="Maroon" title="c23">S1</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:21"><span class="lab"><font color="Green" title="E55">A70</font></span></a>, <a class="txt" href="#E72:21_3"><span class="lab"><font color="Green" title="E105">A127</font></span></a>, <a class="txt" href="#E10:21_3"><span class="lab"><font color="Green" title="E72">A88</font></span></a>, <a class="txt" href="#E115:21_3"><span class="lab"><font color="Green" title="E124">A178</font></span></a>, <a class="ref" href="combgras.html#D4" target="_self" title="COMBGRAS:def.4">Def4</a></span>;<br/>
<a NAME="E117:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c23">S1</font> is  <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="combgras.html#T14" target="_self" title="COMBGRAS:th.14">Th14</a></span>;<br/>
<a NAME="E118:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c23">S1</font> is  <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="combgras.html#D3" target="_self" title="COMBGRAS:def.3">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E119:21_3"/>
<font color="Maroon" title="c3">K</font> is  <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E112:21_3"><span class="lab"><font color="Green" title="E123">A147</font></span></a>, <a class="txt" href="#E115:21_3"><span class="lab"><font color="Green" title="E124">A178</font></span></a>, <a class="txt" href="#E116:21_3"><span class="lab"><font color="Green" title="E125">A179</font></span></a>, <a class="ref" href="combgras.html#D3" target="_self" title="COMBGRAS:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E65:21"/><span class="lab"><font color="Green" title="E67">A180</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/>
<a NAME="E66:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A181</font></span>: 
 <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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T38" title="NAT_1:th.38">NAT_1:38</a></span>;<br/>
<a NAME="E67:21"/>
(  ex <font color="Olive" title="b1">C</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> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Olive" title="b1">C</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="c7">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c8">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">C</font> ) implies <font color="Maroon" title="c3">K</font> is  <a href="combgras.html#V4" title="COMBGRAS:attr.4">TOP</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:21_4"/><span class="lab"><font color="Green" title="E69">A182</font></span>: 
1 <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> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E2:21_4"/><span class="lab"><font color="Green" title="E70">A183</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c8">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a></span>;<br/>
<a NAME="E3:21_4"/><span class="lab"><font color="Green" title="E71">A184</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:21"><span class="lab"><font color="Green" title="E61">A76</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:21_4"/>
 ex <font color="Olive" title="b1">C</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> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Olive" title="b1">C</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="c7">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c8">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">K</font> is  <a href="combgras.html#V4" title="COMBGRAS:attr.4">TOP</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">C</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><span class="kw"> such that </span><br/><a NAME="E6:21_4"/><span class="lab"><font color="Green" title="E72">A185</font></span>: 
<font color="Maroon" title="c10">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">and </span><br/><a NAME="E7:21_4"/><span class="lab"><font color="Green" title="E73">A186</font></span>: 
<font color="Maroon" title="c10">C</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L</font>
 <span class="kw">and </span><br/><a NAME="E8:21_4"/><span class="lab"><font color="Green" title="E74">A187</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">C</font>
 <span class="kw">and </span><br/><a NAME="E9:21_4"/><span class="lab"><font color="Green" title="E75">A188</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">C</font>
 ;<br/>
<a NAME="E10:21_4"/><span class="lab"><font color="Green" title="E76">A189</font></span>: 
<font color="Maroon" title="c10">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E7:21_4"><span class="lab"><font color="Green" title="E73">A186</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E11:21_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:21"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E12:21_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A190</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">C</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:21"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E13:21_4"/>
<font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:21"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E10:21_4"><span class="lab"><font color="Green" title="E76">A189</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E14:21_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A191</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">C</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:21"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E15:21_4"/>
<font color="Maroon" title="c10">C</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>
 
;<br/>
<a NAME="E16:21_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A192</font></span>: 
 ex <font color="Olive" title="b1">C2</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E26">A7</font></span></a></span>;<br/>
<a NAME="E17:21_4"/><span class="kw">then </span>
<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>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E9:21_4"><span class="lab"><font color="Green" title="E75">A188</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E18:21_4"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">C</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21_4"><span class="lab"><font color="Green" title="E78">A191</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E19:21_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A193</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c10">C</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E16:21_4"><span class="lab"><font color="Green" title="E79">A192</font></span></a>, <a class="txt" href="#E2:21_4"><span class="lab"><font color="Green" title="E70">A183</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E20:21_4"/>
<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>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:21"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="txt" href="#E8:21_4"><span class="lab"><font color="Green" title="E74">A187</font></span></a>, <a class="txt" href="#E16:21_4"><span class="lab"><font color="Green" title="E79">A192</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E21:21_4"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">C</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21_4"><span class="lab"><font color="Green" title="E77">A190</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E22:21_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A194</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c10">C</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E16:21_4"><span class="lab"><font color="Green" title="E79">A192</font></span></a>, <a class="txt" href="#E3:21_4"><span class="lab"><font color="Green" title="E71">A184</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E23:21_4"/><span class="lab"><font color="Green" title="E82">A195</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:21"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E61:21"><span class="lab"><font color="Green" title="E63">A78</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E24:21_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A196</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:21"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E54:21"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E47:21"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E48:21"><span class="lab"><font color="Green" title="E52">A67</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E25:21_4"/><span class="lab"><font color="Green" title="E84">A197</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E26:21_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:21"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E10:21_4"><span class="lab"><font color="Green" title="E76">A189</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E27:21_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A198</font></span>: 
 <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="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">C</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:21_4"><span class="lab"><font color="Green" title="E82">A195</font></span></a>, <a class="txt" href="#E24:21_4"><span class="lab"><font color="Green" title="E83">A196</font></span></a>, <a class="txt" href="#E25:21_4"><span class="lab"><font color="Green" title="E84">A197</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E28:21_4"/><span class="lab"><font color="Green" title="E86">A199</font></span>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">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="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; 2 <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> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E59:21"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">T</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E30:21_4"/><span class="lab"><font color="Green" title="E87">A200</font></span>: 
<font color="Maroon" title="c11">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">D</font> where <font color="Olive" title="b1">D</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">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Olive" title="b1">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">L</font> ) </span> } </span> 
 ;<br/>
<a NAME="E31:21_4"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E23:21_4"><span class="lab"><font color="Green" title="E82">A195</font></span></a>, <a class="txt" href="#E3:21_4"><span class="lab"><font color="Green" title="E71">A184</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E32:21_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A201</font></span>: 
( (  <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="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c10">C</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> 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="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c10">C</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 &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="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">C</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E16:21_4"><span class="lab"><font color="Green" title="E79">A192</font></span></a>, <a class="txt" href="#E22:21_4"><span class="lab"><font color="Green" title="E81">A194</font></span></a>, <a class="txt" href="#E28:21_4"><span class="lab"><font color="Green" title="E86">A199</font></span></a>, <a class="txt" href="#E19:21_4"><span class="lab"><font color="Green" title="E80">A193</font></span></a>, <a class="txt" href="#E1:21_4"><span class="lab"><font color="Green" title="E69">A182</font></span></a>, <a class="ref" href="combgras.html#T7" target="_self" title="COMBGRAS:th.7">Th7</a></span>;<br/>
<a NAME="E33:21_4"/><span class="lab"><font color="Green" title="E89">A202</font></span>: 
<font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">D2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">D2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> or <font color="Maroon" title="c12">D2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">T</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21_4_1"/><span class="lab"><font color="Green" title="E90">A203</font></span>: 
<font color="Maroon" title="c12">D2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">and </span><br/><a NAME="E2:21_4_1"/><span class="lab"><font color="Green" title="E91">A204</font></span>: 
 not <font color="Maroon" title="c12">D2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:21_4_1"/>
<font color="Maroon" title="c12">D2</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1"><span class="lab"><font color="Green" title="E90">A203</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">D1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><span class="kw"> such that </span><br/><a NAME="E5:21_4_1"/><span class="lab"><font color="Green" title="E92">A205</font></span>: 
<font color="Maroon" title="c13">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">D2</font>
 <span class="kw">and </span><br/><a NAME="E6:21_4_1"/><span class="lab"><font color="Green" title="E93">A206</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c13">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E26">A7</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">D</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><span class="kw"> such that </span><br/><a NAME="E8:21_4_1"/><span class="lab"><font color="Green" title="E94">A207</font></span>: 
<font color="Maroon" title="c14">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1"><span class="lab"><font color="Green" title="E90">A203</font></span></a>, <a class="txt" href="#E5:21_4_1"><span class="lab"><font color="Green" title="E92">A205</font></span></a></span>;<br/>
<a NAME="E9:21_4_1"/>
 not <font color="Maroon" title="c14">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:21_4"><span class="lab"><font color="Green" title="E87">A200</font></span></a>, <a class="txt" href="#E2:21_4_1"><span class="lab"><font color="Green" title="E91">A204</font></span></a>, <a class="txt" href="#E5:21_4_1"><span class="lab"><font color="Green" title="E92">A205</font></span></a>, <a class="txt" href="#E6:21_4_1"><span class="lab"><font color="Green" title="E93">A206</font></span></a>, <a class="txt" href="#E8:21_4_1"><span class="lab"><font color="Green" title="E94">A207</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:21_4_1"/><span class="lab"><font color="Green" title="E95">A208</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font>
 <span class="kw">and </span><br/><a NAME="E12:21_4_1"/><span class="lab"><font color="Green" title="E96">A209</font></span>: 
 not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E13:21_4_1"/><span class="lab"><font color="Green" title="E97">A210</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T30" title="CARD_1:th.30">CARD_1:30</a></span>;<br/>
<a NAME="E14:21_4_1"/><span class="lab"><font color="Green" title="E98">A211</font></span>: 
<font color="Maroon" title="c14">D</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_4_1"><span class="lab"><font color="Green" title="E93">A206</font></span></a>, <a class="txt" href="#E8:21_4_1"><span class="lab"><font color="Green" title="E94">A207</font></span></a></span>;<br/>
<a NAME="E15:21_4_1"/><span class="lab"><font color="Green" title="E99">A212</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c14">D</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_4_1"><span class="lab"><font color="Green" title="E93">A206</font></span></a>, <a class="txt" href="#E8:21_4_1"><span class="lab"><font color="Green" title="E94">A207</font></span></a></span>;<br/>
<a NAME="E16:21_4_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">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="c14">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21_4_1"><span class="lab"><font color="Green" title="E95">A208</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<a NAME="E17:21_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A213</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">D</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">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="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_4_1"><span class="lab"><font color="Green" title="E93">A206</font></span></a>, <a class="txt" href="#E8:21_4_1"><span class="lab"><font color="Green" title="E94">A207</font></span></a>, <a class="txt" href="#E14:21_4_1"><span class="lab"><font color="Green" title="E98">A211</font></span></a>, <a class="txt" href="#E13:21_4_1"><span class="lab"><font color="Green" title="E97">A210</font></span></a>, <a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">L13</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><span class="kw"> such that </span><br/><a NAME="E19:21_4_1"/><span class="lab"><font color="Green" title="E101">A214</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c14">D</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="c16">L13</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E6:21_4"><span class="lab"><font color="Green" title="E72">A185</font></span></a>, <a class="txt" href="#E1:21_4_1"><span class="lab"><font color="Green" title="E90">A203</font></span></a>, <a class="txt" href="#E5:21_4_1"><span class="lab"><font color="Green" title="E92">A205</font></span></a>, <a class="txt" href="#E8:21_4_1"><span class="lab"><font color="Green" title="E94">A207</font></span></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></span>;<br/>
<a NAME="E20:21_4_1"/>
<font color="Maroon" title="c14">D</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">L13</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:21_4_1"><span class="lab"><font color="Green" title="E101">A214</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E21:21_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A215</font></span>: 
<font color="Maroon" title="c14">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">L13</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E22:21_4_1"/>
<font color="Maroon" title="c16">L13</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/>
<a NAME="E23:21_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A216</font></span>: 
 ex <font color="Olive" title="b1">L23</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">L23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">L13</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">L23</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E33">A14</font></span></a></span>;<br/>
<a NAME="E24:21_4_1"/>
<font color="Maroon" title="c10">C</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">L13</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:21_4_1"><span class="lab"><font color="Green" title="E101">A214</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E25:21_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">L13</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E26:21_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">L13</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:21_4_1"><span class="lab"><font color="Green" title="E102">A215</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E27:21_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A217</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">D</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:21_4_1"><span class="lab"><font color="Green" title="E103">A216</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E28:21_4_1"/><span class="lab"><font color="Green" title="E105">A218</font></span>: 
 not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_4"><span class="lab"><font color="Green" title="E76">A189</font></span></a>, <a class="txt" href="#E12:21_4_1"><span class="lab"><font color="Green" title="E96">A209</font></span></a></span>;<br/>
<a NAME="E29:21_4_1"/><span class="lab"><font color="Green" title="E106">A219</font></span>: 
<font color="Maroon" title="c10">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">D</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font> or <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_4_1_1"/><span class="lab"><font color="Green" title="E107">A220</font></span>: 
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E2:21_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:21_4_1"><span class="lab"><font color="Green" title="E105">A218</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:21_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A221</font></span>: 
 not <font color="Maroon" title="c17">z</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="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E4:21_4_1_1"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1_1"><span class="lab"><font color="Green" title="E107">A220</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:21_4_1_1"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_4_1_1"><span class="lab"><font color="Green" title="E108">A221</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E30:21_4_1"/>
<font color="Maroon" title="c10">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_4"><span class="lab"><font color="Green" title="E76">A189</font></span></a>, <a class="txt" href="#E30:21_4"><span class="lab"><font color="Green" title="E87">A200</font></span></a>, <a class="txt" href="#E2:21_4_1"><span class="lab"><font color="Green" title="E91">A204</font></span></a>, <a class="txt" href="#E5:21_4_1"><span class="lab"><font color="Green" title="E92">A205</font></span></a>, <a class="txt" href="#E6:21_4_1"><span class="lab"><font color="Green" title="E93">A206</font></span></a>, <a class="txt" href="#E8:21_4_1"><span class="lab"><font color="Green" title="E94">A207</font></span></a></span>;<br/>
<a NAME="E31:21_4_1"/><span class="kw">then </span>
<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>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21_4"><span class="lab"><font color="Green" title="E79">A192</font></span></a>, <a class="txt" href="#E6:21_4_1"><span class="lab"><font color="Green" title="E93">A206</font></span></a>, <a class="txt" href="#E8:21_4_1"><span class="lab"><font color="Green" title="E94">A207</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E32:21_4_1"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">D</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:21_4_1"><span class="lab"><font color="Green" title="E104">A217</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E33:21_4_1"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E16:21_4"><span class="lab"><font color="Green" title="E79">A192</font></span></a>, <a class="txt" href="#E15:21_4_1"><span class="lab"><font color="Green" title="E99">A212</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E34:21_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A222</font></span>: 
<font color="Maroon" title="c10">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">D</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21_4_1"><span class="lab"><font color="Green" title="E98">A211</font></span></a>, <a class="txt" href="#E17:21_4_1"><span class="lab"><font color="Green" title="E100">A213</font></span></a>, <a class="txt" href="#E29:21_4_1"><span class="lab"><font color="Green" title="E106">A219</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">L12</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><span class="kw"> such that </span><br/><a NAME="E36:21_4_1"/><span class="lab"><font color="Green" title="E108">A223</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c14">D</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="c17">L12</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E40">A55</font></span></a>, <a class="txt" href="#E40:21"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E43:21"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E1:21_4_1"><span class="lab"><font color="Green" title="E90">A203</font></span></a>, <a class="txt" href="#E5:21_4_1"><span class="lab"><font color="Green" title="E92">A205</font></span></a>, <a class="txt" href="#E8:21_4_1"><span class="lab"><font color="Green" title="E94">A207</font></span></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">L11</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><span class="kw"> such that </span><br/><a NAME="E38:21_4_1"/><span class="lab"><font color="Green" title="E109">A224</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c14">D</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="c18">L11</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E40">A55</font></span></a>, <a class="txt" href="#E36:21"><span class="lab"><font color="Green" title="E44">A59</font></span></a>, <a class="txt" href="#E39:21"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E1:21_4_1"><span class="lab"><font color="Green" title="E90">A203</font></span></a>, <a class="txt" href="#E5:21_4_1"><span class="lab"><font color="Green" title="E92">A205</font></span></a>, <a class="txt" href="#E8:21_4_1"><span class="lab"><font color="Green" title="E94">A207</font></span></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></span>;<br/>
<a NAME="E39:21_4_1"/>
<font color="Maroon" title="c14">D</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">L11</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:21_4_1"><span class="lab"><font color="Green" title="E109">A224</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E40:21_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A225</font></span>: 
<font color="Maroon" title="c14">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">L11</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E41:21_4_1"/>
<font color="Maroon" title="c18">L11</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/>
<a NAME="E42:21_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A226</font></span>: 
 ex <font color="Olive" title="b1">L21</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">L21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">L11</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">L21</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E33">A14</font></span></a></span>;<br/>
<a NAME="E43:21_4_1"/>
<font color="Maroon" title="c7">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">L11</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:21_4_1"><span class="lab"><font color="Green" title="E109">A224</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E44:21_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">L11</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E45:21_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">L11</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:21_4_1"><span class="lab"><font color="Green" title="E110">A225</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E46:21_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A227</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">D</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:21_4_1"><span class="lab"><font color="Green" title="E111">A226</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E47:21_4_1"/><span class="lab"><font color="Green" title="E113">A228</font></span>: 
 not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:21"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E12:21_4_1"><span class="lab"><font color="Green" title="E96">A209</font></span></a></span>;<br/>
<a NAME="E48:21_4_1"/><span class="lab"><font color="Green" title="E114">A229</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">D</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font> or <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_4_1_2"/><span class="lab"><font color="Green" title="E115">A230</font></span>: 
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E2:21_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:21_4_1"><span class="lab"><font color="Green" title="E113">A228</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:21_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A231</font></span>: 
 not <font color="Maroon" title="c19">z</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="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E4:21_4_1_2"/>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1_2"><span class="lab"><font color="Green" title="E115">A230</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:21_4_1_2"/>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_4_1_2"><span class="lab"><font color="Green" title="E116">A231</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E49:21_4_1"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:21"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E30:21_4"><span class="lab"><font color="Green" title="E87">A200</font></span></a>, <a class="txt" href="#E2:21_4_1"><span class="lab"><font color="Green" title="E91">A204</font></span></a>, <a class="txt" href="#E5:21_4_1"><span class="lab"><font color="Green" title="E92">A205</font></span></a>, <a class="txt" href="#E6:21_4_1"><span class="lab"><font color="Green" title="E93">A206</font></span></a>, <a class="txt" href="#E8:21_4_1"><span class="lab"><font color="Green" title="E94">A207</font></span></a></span>;<br/>
<a NAME="E50:21_4_1"/><span class="kw">then </span>
<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>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:21"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="txt" href="#E6:21_4_1"><span class="lab"><font color="Green" title="E93">A206</font></span></a>, <a class="txt" href="#E8:21_4_1"><span class="lab"><font color="Green" title="E94">A207</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E51:21_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A232</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">D</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:21_4_1"><span class="lab"><font color="Green" title="E112">A227</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E52:21_4_1"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E59:21"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="txt" href="#E15:21_4_1"><span class="lab"><font color="Green" title="E99">A212</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E53:21_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A233</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">D</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21_4_1"><span class="lab"><font color="Green" title="E98">A211</font></span></a>, <a class="txt" href="#E17:21_4_1"><span class="lab"><font color="Green" title="E100">A213</font></span></a>, <a class="txt" href="#E48:21_4_1"><span class="lab"><font color="Green" title="E114">A229</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a></span>;<br/>
<a NAME="E54:21_4_1"/>
<font color="Maroon" title="c14">D</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">L12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:21_4_1"><span class="lab"><font color="Green" title="E108">A223</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E55:21_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A234</font></span>: 
<font color="Maroon" title="c14">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">L12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E56:21_4_1"/>
<font color="Maroon" title="c17">L12</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/>
<a NAME="E57:21_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A235</font></span>: 
 ex <font color="Olive" title="b1">L22</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">L22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">L12</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">L22</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E33">A14</font></span></a></span>;<br/>
<a NAME="E58:21_4_1"/>
<font color="Maroon" title="c8">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">L12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:21_4_1"><span class="lab"><font color="Green" title="E108">A223</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E59:21_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">L12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E60:21_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">L12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:21_4_1"><span class="lab"><font color="Green" title="E117">A234</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E61:21_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A236</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">D</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:21_4_1"><span class="lab"><font color="Green" title="E118">A235</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E62:21_4_1"/><span class="lab"><font color="Green" title="E120">A237</font></span>: 
 not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:21"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E12:21_4_1"><span class="lab"><font color="Green" title="E96">A209</font></span></a></span>;<br/>
<a NAME="E63:21_4_1"/><span class="lab"><font color="Green" title="E121">A238</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">D</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font> or <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_4_1_3"/><span class="lab"><font color="Green" title="E122">A239</font></span>: 
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E2:21_4_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:21_4_1"><span class="lab"><font color="Green" title="E120">A237</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:21_4_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A240</font></span>: 
 not <font color="Maroon" title="c19">z</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="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E4:21_4_1_3"/>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1_3"><span class="lab"><font color="Green" title="E122">A239</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:21_4_1_3"/>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_4_1_3"><span class="lab"><font color="Green" title="E123">A240</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E64:21_4_1"/>
<font color="Maroon" title="c8">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:21"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E30:21_4"><span class="lab"><font color="Green" title="E87">A200</font></span></a>, <a class="txt" href="#E2:21_4_1"><span class="lab"><font color="Green" title="E91">A204</font></span></a>, <a class="txt" href="#E5:21_4_1"><span class="lab"><font color="Green" title="E92">A205</font></span></a>, <a class="txt" href="#E6:21_4_1"><span class="lab"><font color="Green" title="E93">A206</font></span></a>, <a class="txt" href="#E8:21_4_1"><span class="lab"><font color="Green" title="E94">A207</font></span></a></span>;<br/>
<a NAME="E65:21_4_1"/><span class="kw">then </span>
<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>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E6:21_4_1"><span class="lab"><font color="Green" title="E93">A206</font></span></a>, <a class="txt" href="#E8:21_4_1"><span class="lab"><font color="Green" title="E94">A207</font></span></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a></span>;<br/>
<a NAME="E66:21_4_1"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">D</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E61:21_4_1"><span class="lab"><font color="Green" title="E119">A236</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E67:21_4_1"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E57:21"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E15:21_4_1"><span class="lab"><font color="Green" title="E99">A212</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E68:21_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">D</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21_4_1"><span class="lab"><font color="Green" title="E98">A211</font></span></a>, <a class="txt" href="#E17:21_4_1"><span class="lab"><font color="Green" title="E100">A213</font></span></a>, <a class="txt" href="#E63:21_4_1"><span class="lab"><font color="Green" title="E121">A238</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a></span>;<br/>
<a NAME="E69:21_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</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="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:21_4_1"><span class="lab"><font color="Green" title="E116">A233</font></span></a></span>;<br/>
<a NAME="E70:21_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">D</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>;<br/>
<a NAME="E71:21_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>;<br/>
<a NAME="E72:21_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</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="c14">D</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>;<br/>
<a NAME="E73:21_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</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="c10">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:21_4_1"><span class="lab"><font color="Green" title="E116">A233</font></span></a>, <a class="txt" href="#E34:21_4_1"><span class="lab"><font color="Green" title="E107">A222</font></span></a></span>;<br/>
<a NAME="E74:21_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">D</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c10">C</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>;<br/>
<a NAME="E75:21_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c10">C</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>;<br/>
<a NAME="E76:21_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c10">C</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="c14">D</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>;<br/>
<a NAME="E77:21_4_1"/><span class="kw">then </span>
 <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"><span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c10">C</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">D</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E59:21"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="txt" href="#E15:21_4_1"><span class="lab"><font color="Green" title="E99">A212</font></span></a>, <a class="txt" href="#E51:21_4_1"><span class="lab"><font color="Green" title="E115">A232</font></span></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a></span>;<br/>
<a NAME="E78:21_4_1"/><span class="kw">then </span>
<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> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:21_4"><span class="lab"><font color="Green" title="E85">A198</font></span></a>, <a class="txt" href="#E32:21_4"><span class="lab"><font color="Green" title="E88">A201</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E79:21_4_1"/><span class="kw">then </span>
<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> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E65:21"><span class="lab"><font color="Green" title="E67">A180</font></span></a>, <a class="txt" href="#E66:21"><span class="lab"><font color="Green" title="E68">A181</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E80:21_4_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E34:21_4"/>
<font color="Maroon" title="c11">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">T</font> or <font color="Maroon" title="c12">e</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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_4_2"/>
<font color="Maroon" title="c12">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">e</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></span><br/>

<a NAME="E2:21_4_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">E</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Maroon" title="c12">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">E</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Olive" title="b1">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:21_4"><span class="lab"><font color="Green" title="E87">A200</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21_4_2"/>
<font color="Maroon" title="c12">e</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E26">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">T1</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><span class="kw"> such that </span><br/><a NAME="E36:21_4"/><span class="lab"><font color="Green" title="E90">A241</font></span>: 
<font color="Maroon" title="c12">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">T</font>
 ;<br/>
<a NAME="E37:21_4"/><span class="lab"><font color="Green" title="E91">A242</font></span>: 
<font color="Maroon" title="c12">T1</font> is  <a href="combgras.html#V4" title="COMBGRAS:attr.4">TOP</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:21"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E30:21_4"><span class="lab"><font color="Green" title="E87">A200</font></span></a>, <a class="txt" href="#E36:21_4"><span class="lab"><font color="Green" title="E90">A241</font></span></a>, <a class="ref" href="combgras.html#D5" target="_self" title="COMBGRAS:def.5">Def5</a></span>;<br/>
<a NAME="E38:21_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">T1</font> is  <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="combgras.html#T14" target="_self" title="COMBGRAS:th.14">Th14</a></span>;<br/>
<a NAME="E39:21_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">T1</font> is  <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="combgras.html#D3" target="_self" title="COMBGRAS:def.3">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E40:21_4"/>
<font color="Maroon" title="c3">K</font> is  <a href="combgras.html#V4" title="COMBGRAS:attr.4">TOP</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E33:21_4"><span class="lab"><font color="Green" title="E89">A202</font></span></a>, <a class="txt" href="#E36:21_4"><span class="lab"><font color="Green" title="E90">A241</font></span></a>, <a class="txt" href="#E37:21_4"><span class="lab"><font color="Green" title="E91">A242</font></span></a>, <a class="ref" href="combgras.html#D3" target="_self" title="COMBGRAS:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E68:21"/>
( <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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:21"><span class="lab"><font color="Green" title="E66">A81</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
