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<b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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">F</font> being   <a href="combgras.html#L1" title="COMBGRAS:struct.1">IncProjMap</a> of  <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>, <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>  st <font color="Olive" title="b2">F</font> is <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> holds <br/>for <font color="Olive" title="b3">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>  st <font color="Olive" title="b3">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> holds <br/>( <font color="Olive" title="b2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Olive" title="b3">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> &amp; <font color="Olive" title="b2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Olive" title="b3">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> )</span><br/><b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">F</font> being   <a href="combgras.html#L1" title="COMBGRAS:struct.1">IncProjMap</a> of  <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>, <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>  st <font color="Olive" title="b1">F</font> is <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> 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="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b2">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> holds <br/>( <font color="Olive" title="b1">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Olive" title="b2">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> &amp; <font color="Olive" title="b1">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Olive" title="b2">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> ) )</span><br/>



<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E29">A1</font></i>: 
( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being   <a href="combgras.html#L1" title="COMBGRAS:struct.1">IncProjMap</a> of  <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>, <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>  st <font color="Olive" title="b1">F</font> is <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> 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="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b2">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> holds <br/>( <font color="Olive" title="b1">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Olive" title="b2">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> &amp; <font color="Olive" title="b1">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Olive" title="b2">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> )</span><br/>

<b>let </b><font color="Maroon" title="c3">F</font> be    <a href="combgras.html#L1" title="COMBGRAS:struct.1">IncProjMap</a> of  <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>, <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> is <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> 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>  st <font color="Olive" title="b1">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> holds <br/>( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Olive" title="b1">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Olive" title="b1">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> ) )</span><br/>

<b>assume </b><a NAME="E2:38"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c3">F</font> is <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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>  st <font color="Olive" title="b1">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> holds <br/>( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Olive" title="b1">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Olive" title="b1">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> )</span><br/>

<b>let </b><font color="Maroon" title="c4">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> implies ( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> ) )</span><br/>

<a NAME="E3:38"/><i><font color="Green" title="E31">A3</font></i>: 
( <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="#E1:38"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:38"/><b>then </b>
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/>
<a NAME="E5:38"/><b>then </b><i><font color="Green" title="E32">A4</font></i>: 
( 2 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="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> &amp; 2 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; 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 &amp; <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>  &amp; <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>  )
 
<b>by </b><i><a class="txt" href="#E1:38"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E3:38"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:38"/><b>then </b><i><font color="Green" title="E33">A5</font></i>: 
( 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> 1 &amp; 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> &amp; 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> 1 &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="#E3:38"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a>, <a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a>, <a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/>
<a NAME="E7:38"/><i><font color="Green" title="E34">A6</font></i>: 
the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E5:38"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></i>;<br/>
<a NAME="E8:38"/><i><font color="Green" title="E35">A7</font></i>: 
the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E5:38"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></i>;<br/>
<b>assume </b><a NAME="E9:38"/><i><font color="Green" title="E36">A8</font></i>: 
<font color="Maroon" title="c4">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> )</span><br/>

<a NAME="E10:38"/><b>then </b>
<font color="Maroon" title="c4">K</font> is <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a>
 
<b>by </b><i><a class="txt" href="#E1:38"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="combgras.html#T14" target="_self" title="COMBGRAS:th.14">Th14</a></i>;<br/>
<a NAME="E11:38"/><b>then </b><i><font color="Green" title="E37">A9</font></i>: 
( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font> is <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font> is <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a> &amp; <font color="Maroon" title="c4">K</font> is <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> )
 
<b>by </b><i><a class="txt" href="#E2:38"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="combgras.html#D3" target="_self" title="COMBGRAS:def.3">Def3</a>, <a class="ref" href="combgras.html#T22" target="_self" title="COMBGRAS:th.22">Th22</a></i>;<br/>
<a NAME="E12:38"/><i><font color="Green" title="E38">A10</font></i>: 
not <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font> is <a href="combgras.html#V4" title="COMBGRAS:attr.4">TOP</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font> is <a href="combgras.html#V4" title="COMBGRAS:attr.4">TOP</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c5">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E3:38_1"/><i><font color="Green" title="E39">A11</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> : (  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="combgras.html#D5" target="_self" title="COMBGRAS:def.5">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:38_1"/><i><font color="Green" title="E40">A12</font></i>: 
( <font color="Maroon" title="c6">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="#E6:38"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="#E3:38_1"><i><font color="Green" title="E39">A11</font></i></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></i>;<br/>
<a NAME="E6:38_1"/><i><font color="Green" title="E41">A13</font></i>: 
( <font color="Maroon" title="c6">X1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c5">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="txt" href="#E3:38_1"><i><font color="Green" title="E39">A11</font></i></a>, <a class="txt" href="#E5:38_1"><i><font color="Green" title="E40">A12</font></i></a></i>;<br/>
<a NAME="E7:38_1"/><b>then </b><i><font color="Green" title="E42">A14</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</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> 1
 
<b>by </b><i><a class="txt" href="#E3:38_1"><i><font color="Green" title="E39">A11</font></i></a>, <a class="txt" href="#E5:38_1"><i><font color="Green" title="E40">A12</font></i></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">X2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:38_1"/><i><font color="Green" title="E43">A15</font></i>: 
( <font color="Maroon" title="c7">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="#E6:38"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></i>;<br/>
<a NAME="E10:38_1"/>
( <font color="Maroon" title="c7">X2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="txt" href="#E7:38_1"><i><font color="Green" title="E42">A14</font></i></a>, <a class="txt" href="#E9:38_1"><i><font color="Green" title="E43">A15</font></i></a></i>;<br/>
<a NAME="E11:38_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="txt" href="#E7:38_1"><i><font color="Green" title="E42">A14</font></i></a>, <a class="txt" href="#E9:38_1"><i><font color="Green" title="E43">A15</font></i></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">X3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:38_1"/><i><font color="Green" title="E44">A17</font></i>: 
( <font color="Maroon" title="c8">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c8">X3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="#E6:38"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></i>;<br/>
<a NAME="E14:38_1"/>
( <font color="Maroon" title="c8">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">X2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">X1</font></span>)</span> &amp; the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> &amp; the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> )
 
<b>by </b><i><a class="txt" href="#E2:38"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="#E13:38_1"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a>, <a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a></i>;<br/>
<a NAME="E15:38_1"/><b>then </b><i><font color="Green" title="E45">A18</font></i>: 
( <font color="Maroon" title="c8">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c7">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c8">X3</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c8">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> &amp; the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; <font color="Maroon" title="c3">F</font> is <a href="combgras.html#V6" title="COMBGRAS:attr.6">incidence_preserving</a> &amp; the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="#E2:38"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="#E9:38_1"><i><font color="Green" title="E43">A15</font></i></a>, <a class="txt" href="#E13:38_1"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a>, <a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a>, <a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></i>;<br/>
<a NAME="E16:38_1"/><b>then </b><i><font color="Green" title="E46">A19</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</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> 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="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</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> 1 &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c8">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> &amp; <font color="Maroon" title="c8">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> &amp; <font color="Maroon" title="c6">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">X2</font> &amp; <font color="Maroon" title="c7">X2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">X3</font> &amp; <font color="Maroon" title="c6">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">X3</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> &amp; 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>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; 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>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> &amp; 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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; 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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> )
 
<b>by </b><i><a class="txt" href="#E3:38_1"><i><font color="Green" title="E39">A11</font></i></a>, <a class="txt" href="#E6:38_1"><i><font color="Green" title="E41">A13</font></i></a>, <a class="txt" href="#E9:38_1"><i><font color="Green" title="E43">A15</font></i></a>, <a class="txt" href="#E13:38_1"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a>, <a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a>, <a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></i>;<br/>
<a NAME="E17:38_1"/><b>then </b><i><font color="Green" title="E47">A20</font></i>: 
( <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</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> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</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> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</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> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E7:38"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="#E8:38"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="#E3:38_1"><i><font color="Green" title="E39">A11</font></i></a>, <a class="txt" href="#E7:38_1"><i><font color="Green" title="E42">A14</font></i></a></i>;<br/>
<a NAME="E18:38_1"/><i><font color="Green" title="E48">A21</font></i>: 
( <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1_1"/>
( <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> or <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</font> or <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:38_1_1"/><b>then </b>
( <font color="Maroon" title="c7">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> or <font color="Maroon" title="c8">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</font> or <font color="Maroon" title="c8">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</font> )
 
<b>by </b><i><a class="txt" href="#E9:38_1"><i><font color="Green" title="E43">A15</font></i></a>, <a class="txt" href="#E13:38_1"><i><font color="Green" title="E44">A17</font></i></a>, <a class="txt" href="#E15:38_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></i>;<br/>
<a NAME="E3:38_1_1"/><b>then </b>
( <font color="Maroon" title="c7">X2</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">X3</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">X3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T38" title="XBOOLE_1:th.38">XBOOLE_1:38</a></i>;<br/>
<b>hence </b><a NAME="E4:38_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E9:38_1"><i><font color="Green" title="E43">A15</font></i></a>, <a class="txt" href="#E13:38_1"><i><font color="Green" title="E44">A17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c9">B1</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E20:38_1"/><i><font color="Green" title="E49">A22</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B1</font>
 <b>by </b><i><a class="txt" href="#E17:38_1"><i><font color="Green" title="E47">A20</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">B2</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E22:38_1"/><i><font color="Green" title="E50">A23</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B2</font>
 <b>by </b><i><a class="txt" href="#E17:38_1"><i><font color="Green" title="E47">A20</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">B3</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E24:38_1"/><i><font color="Green" title="E51">A24</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">B3</font>
 <b>by </b><i><a class="txt" href="#E17:38_1"><i><font color="Green" title="E47">A20</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">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><b> such that </b><br/><a NAME="E26:38_1"/><i><font color="Green" title="E52">A25</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">L2</font>
 <b>by </b><i><a class="txt" href="#E17:38_1"><i><font color="Green" title="E47">A20</font></i></a></i>;<br/>
<a NAME="E27:38_1"/><i><font color="Green" title="E53">A26</font></i>: 
( <font color="Maroon" title="c9">B1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">L2</font> &amp; <font color="Maroon" title="c10">B2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">L2</font> &amp; <font color="Maroon" title="c11">B3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">L2</font> &amp; <font color="Maroon" title="c9">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c10">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c11">B3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c12">L2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> )
 
<b>by </b><i><a class="txt" href="#E5:38"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E16:38_1"><i><font color="Green" title="E46">A19</font></i></a>, <a class="txt" href="#E20:38_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="#E22:38_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="txt" href="#E24:38_1"><i><font color="Green" title="E51">A24</font></i></a>, <a class="txt" href="#E26:38_1"><i><font color="Green" title="E52">A25</font></i></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">a1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E29:38_1"/><i><font color="Green" title="E54">A27</font></i>: 
( <font color="Maroon" title="c13">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c9">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a1</font> )
 <b>by </b><i><a class="txt" href="#E16:38_1"><i><font color="Green" title="E46">A19</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E31:38_1"/><i><font color="Green" title="E55">A28</font></i>: 
( <font color="Maroon" title="c14">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c10">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a2</font> )
 <b>by </b><i><a class="txt" href="#E16:38_1"><i><font color="Green" title="E46">A19</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">a3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E33:38_1"/><i><font color="Green" title="E56">A29</font></i>: 
( <font color="Maroon" title="c15">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c11">B3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a3</font> )
 <b>by </b><i><a class="txt" href="#E16:38_1"><i><font color="Green" title="E46">A19</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">l1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E35:38_1"/><i><font color="Green" title="E57">A30</font></i>: 
( <font color="Maroon" title="c16">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c12">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">l1</font> )
 <b>by </b><i><a class="txt" href="#E16:38_1"><i><font color="Green" title="E46">A19</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">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><b> such that </b><br/><a NAME="E37:38_1"/><i><font color="Green" title="E58">A31</font></i>: 
<font color="Maroon" title="c13">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">A1</font>
 <b>by </b><i><a class="txt" href="#E29:38_1"><i><font color="Green" title="E54">A27</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">A2</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E39:38_1"/><i><font color="Green" title="E59">A32</font></i>: 
<font color="Maroon" title="c14">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">A2</font>
 <b>by </b><i><a class="txt" href="#E31:38_1"><i><font color="Green" title="E55">A28</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">A3</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E41:38_1"/><i><font color="Green" title="E60">A33</font></i>: 
<font color="Maroon" title="c15">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">A3</font>
 <b>by </b><i><a class="txt" href="#E33:38_1"><i><font color="Green" title="E56">A29</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">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><b> such that </b><br/><a NAME="E43:38_1"/><i><font color="Green" title="E61">A34</font></i>: 
<font color="Maroon" title="c16">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">L1</font>
 <b>by </b><i><a class="txt" href="#E35:38_1"><i><font color="Green" title="E57">A30</font></i></a></i>;<br/>
<a NAME="E44:38_1"/><i><font color="Green" title="E62">A35</font></i>: 
( <font color="Maroon" title="c9">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c17">A1</font> &amp; <font color="Maroon" title="c10">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c18">A2</font> &amp; <font color="Maroon" title="c11">B3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c19">A3</font> &amp; <font color="Maroon" title="c12">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c20">L1</font> )
 
<b>by </b><i><a class="txt" href="#E29:38_1"><i><font color="Green" title="E54">A27</font></i></a>, <a class="txt" href="#E31:38_1"><i><font color="Green" title="E55">A28</font></i></a>, <a class="txt" href="#E33:38_1"><i><font color="Green" title="E56">A29</font></i></a>, <a class="txt" href="#E35:38_1"><i><font color="Green" title="E57">A30</font></i></a>, <a class="txt" href="#E37:38_1"><i><font color="Green" title="E58">A31</font></i></a>, <a class="txt" href="#E39:38_1"><i><font color="Green" title="E59">A32</font></i></a>, <a class="txt" href="#E41:38_1"><i><font color="Green" title="E60">A33</font></i></a>, <a class="txt" href="#E43:38_1"><i><font color="Green" title="E61">A34</font></i></a></i>;<br/>
<a NAME="E45:38_1"/><b>then </b>
( <font color="Maroon" title="c17">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">L1</font> &amp; <font color="Maroon" title="c18">A2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">L1</font> &amp; <font color="Maroon" title="c19">A3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">L1</font> )
 
<b>by </b><i><a class="txt" href="#E15:38_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="#E27:38_1"><i><font color="Green" title="E53">A26</font></i></a>, <a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">Def8</a></i>;<br/>
<a NAME="E46:38_1"/><b>then </b><i><font color="Green" title="E63">A36</font></i>: 
( <font color="Maroon" title="c17">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">L1</font> &amp; <font color="Maroon" title="c18">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">L1</font> &amp; <font color="Maroon" title="c19">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">L1</font> &amp; <font color="Maroon" title="c17">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">A2</font> &amp; <font color="Maroon" title="c18">A2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">A3</font> &amp; <font color="Maroon" title="c17">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">A3</font> &amp; <font color="Maroon" title="c9">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c10">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c11">B3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font> )
 
<b>by </b><i><a class="txt" href="#E5:38"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E3:38_1"><i><font color="Green" title="E39">A11</font></i></a>, <a class="txt" href="#E7:38_1"><i><font color="Green" title="E42">A14</font></i></a>, <a class="txt" href="#E16:38_1"><i><font color="Green" title="E46">A19</font></i></a>, <a class="txt" href="#E18:38_1"><i><font color="Green" title="E48">A21</font></i></a>, <a class="txt" href="#E20:38_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="#E22:38_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="txt" href="#E24:38_1"><i><font color="Green" title="E51">A24</font></i></a>, <a class="txt" href="#E29:38_1"><i><font color="Green" title="E54">A27</font></i></a>, <a class="txt" href="#E31:38_1"><i><font color="Green" title="E55">A28</font></i></a>, <a class="txt" href="#E33:38_1"><i><font color="Green" title="E56">A29</font></i></a>, <a class="txt" href="#E37:38_1"><i><font color="Green" title="E58">A31</font></i></a>, <a class="txt" href="#E39:38_1"><i><font color="Green" title="E59">A32</font></i></a>, <a class="txt" href="#E41:38_1"><i><font color="Green" title="E60">A33</font></i></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">C1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E48:38_1"/><i><font color="Green" title="E64">A37</font></i>: 
( <font color="Maroon" title="c21">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c21">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c9">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">C1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">C2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E50:38_1"/><i><font color="Green" title="E65">A38</font></i>: 
( <font color="Maroon" title="c22">C2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c22">C2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c10">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">C2</font> )
 <b>by </b><i><a class="txt" href="#E46:38_1"><i><font color="Green" title="E63">A36</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c23">C3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E52:38_1"/><i><font color="Green" title="E66">A39</font></i>: 
( <font color="Maroon" title="c23">C3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c23">C3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c11">B3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">C3</font> )
 <b>by </b><i><a class="txt" href="#E46:38_1"><i><font color="Green" title="E63">A36</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E53:38_1"/><i><font color="Green" title="E67">A40</font></i>: 
( <font color="Maroon" title="c17">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c18">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c19">A3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> )
 
<b>by </b><i><a class="txt" href="#E15:38_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="#E29:38_1"><i><font color="Green" title="E54">A27</font></i></a>, <a class="txt" href="#E31:38_1"><i><font color="Green" title="E55">A28</font></i></a>, <a class="txt" href="#E33:38_1"><i><font color="Green" title="E56">A29</font></i></a>, <a class="txt" href="#E37:38_1"><i><font color="Green" title="E58">A31</font></i></a>, <a class="txt" href="#E39:38_1"><i><font color="Green" title="E59">A32</font></i></a>, <a class="txt" href="#E41:38_1"><i><font color="Green" title="E60">A33</font></i></a>, <a class="txt" href="#E48:38_1"><i><font color="Green" title="E64">A37</font></i></a>, <a class="txt" href="#E50:38_1"><i><font color="Green" title="E65">A38</font></i></a>, <a class="txt" href="#E52:38_1"><i><font color="Green" title="E66">A39</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c24">L3a</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E55:38_1"/><i><font color="Green" title="E68">A41</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c17">A1</font>,<font color="Maroon" title="c18">A2</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="c24">L3a</font>
 <b>by </b><i><a class="txt" href="#E11:38"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">L3b</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E57:38_1"/><i><font color="Green" title="E69">A42</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">A2</font>,<font color="Maroon" title="c19">A3</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="c25">L3b</font>
 <b>by </b><i><a class="txt" href="#E11:38"><i><font color="Green" title="E37">A9</font></i></a>, <a class="txt" href="#E53:38_1"><i><font color="Green" title="E67">A40</font></i></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">S</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E59:38_1"/><i><font color="Green" title="E70">A43</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c26">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; <font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> : (  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c26">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="#E9:38"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="combgras.html#D4" target="_self" title="COMBGRAS:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c27">A11</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E61:38_1"/><i><font color="Green" title="E71">A44</font></i>: 
( <font color="Maroon" title="c17">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">A11</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c27">A11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c26">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">A11</font> )
 <b>by </b><i><a class="txt" href="#E53:38_1"><i><font color="Green" title="E67">A40</font></i></a>, <a class="txt" href="#E59:38_1"><i><font color="Green" title="E70">A43</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c28">A12</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E63:38_1"/><i><font color="Green" title="E72">A45</font></i>: 
( <font color="Maroon" title="c18">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">A12</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c28">A12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c26">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c28">A12</font> )
 <b>by </b><i><a class="txt" href="#E53:38_1"><i><font color="Green" title="E67">A40</font></i></a>, <a class="txt" href="#E59:38_1"><i><font color="Green" title="E70">A43</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">A13</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E65:38_1"/><i><font color="Green" title="E73">A46</font></i>: 
( <font color="Maroon" title="c19">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">A13</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c29">A13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c26">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c29">A13</font> )
 <b>by </b><i><a class="txt" href="#E53:38_1"><i><font color="Green" title="E67">A40</font></i></a>, <a class="txt" href="#E59:38_1"><i><font color="Green" title="E70">A43</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c30">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E67:38_1"/><i><font color="Green" title="E74">A47</font></i>: 
<font color="Maroon" title="c30">m</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
 <b>by </b><i><a class="txt" href="#E5:38"><i><font color="Green" title="E32">A4</font></i></a></i>;<br/>
<a NAME="E68:38_1"/>
<font color="Maroon" title="c20">L1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c31">l12</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E70:38_1"/><i><font color="Green" title="E75">A48</font></i>: 
( <font color="Maroon" title="c20">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">l12</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c31">l12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="#E8:38"><i><font color="Green" title="E35">A7</font></i></a></i>;<br/>
<a NAME="E71:38_1"/>
( <font color="Maroon" title="c26">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">A2</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">A2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">L1</font> &amp; <font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c19">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">L1</font> &amp; <font color="Maroon" title="c18">A2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c19">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">L1</font> &amp; 2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> )
 
<b>by </b><i><a class="txt" href="#E1:38"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E46:38_1"><i><font color="Green" title="E63">A36</font></i></a>, <a class="txt" href="#E61:38_1"><i><font color="Green" title="E71">A44</font></i></a>, <a class="txt" href="#E63:38_1"><i><font color="Green" title="E72">A45</font></i></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a>, <a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E72:38_1"/><b>then </b>
( <font color="Maroon" title="c26">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c19">A3</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">A2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c19">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">A2</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">A2</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 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c19">A3</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 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">A2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c19">A3</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 &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">A2</font></span>)</span> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c19">A3</font></span>)</span> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">A2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c19">A3</font></span>)</span> &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">k</font> )
 
<b>by </b><i><a class="txt" href="#E6:38"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="#E46:38_1"><i><font color="Green" title="E63">A36</font></i></a>, <a class="txt" href="#E61:38_1"><i><font color="Green" title="E71">A44</font></i></a>, <a class="txt" href="#E63:38_1"><i><font color="Green" title="E72">A45</font></i></a>, <a class="txt" href="#E65:38_1"><i><font color="Green" title="E73">A46</font></i></a>, <a class="txt" href="#E70:38_1"><i><font color="Green" title="E75">A48</font></i></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</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#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E73:38_1"/><b>then </b><i><font color="Green" title="E76">A49</font></i>: 
( <font color="Maroon" title="c30">m</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c19">A3</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">A2</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c30">m</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">A2</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 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">A2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c19">A3</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 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c19">A3</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 &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="c17">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c19">A3</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">A2</font></span>)</span> &amp; ( 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> or <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 
<b>by </b><i><a class="txt" href="#E59:38_1"><i><font color="Green" title="E70">A43</font></i></a>, <a class="txt" href="#E67:38_1"><i><font color="Green" title="E74">A47</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="ordinal1.html#T13" title="ORDINAL1:th.13">ORDINAL1:13</a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E74:38_1"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c19">A3</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c30">m</font> &amp; ( 3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> or <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 
<b>by </b><i><a class="txt" href="#E6:38"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E75:38_1"/><b>then </b><i><font color="Green" title="E77">A50</font></i>: 
( <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c17">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c19">A3</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c17">A1</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="c18">A2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c19">A3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c19">A3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c18">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c19">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )
 
<b>by </b><i><a class="txt" href="#E61:38_1"><i><font color="Green" title="E71">A44</font></i></a>, <a class="txt" href="#E63:38_1"><i><font color="Green" title="E72">A45</font></i></a>, <a class="txt" href="#E65:38_1"><i><font color="Green" title="E73">A46</font></i></a>, <a class="txt" href="#E67:38_1"><i><font color="Green" title="E74">A47</font></i></a>, <a class="txt" href="#E73:38_1"><i><font color="Green" title="E76">A49</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E76:38_1"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">A2</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> <font color="Maroon" title="c17">A1</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;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">A2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c19">A3</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"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c19">A3</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> <font color="Maroon" title="c18">A2</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;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c19">A3</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 &amp; 2 <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 &amp; <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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; <font color="Maroon" title="c17">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c24">L3a</font> &amp; <font color="Maroon" title="c18">A2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">L3b</font> &amp; <font color="Maroon" title="c19">A3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">L3b</font> )
 
<b>by </b><i><a class="txt" href="#E1:38"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E3:38"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="#E5:38"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E55:38_1"><i><font color="Green" title="E68">A41</font></i></a>, <a class="txt" href="#E57:38_1"><i><font color="Green" title="E69">A42</font></i></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E77:38_1"/><b>then </b><i><font color="Green" title="E78">A51</font></i>: 
(  <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="c17">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">A2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c19">A3</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 &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="c17">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c19">A3</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; <font color="Maroon" title="c17">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c24">L3a</font> &amp; <font color="Maroon" title="c18">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">L3b</font> &amp; <font color="Maroon" title="c19">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">L3b</font> )
 
<b>by </b><i><a class="txt" href="#E5:38"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E75:38_1"><i><font color="Green" title="E77">A50</font></i></a>, <a class="ref" href="combgras.html#T7" target="_self" title="COMBGRAS:th.7">Th7</a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></i>;<br/>
<a NAME="E78:38_1"/>
<font color="Maroon" title="c25">L3b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c32">l13b</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E80:38_1"/><i><font color="Green" title="E79">A52</font></i>: 
( <font color="Maroon" title="c25">L3b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">l13b</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c32">l13b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="#E8:38"><i><font color="Green" title="E35">A7</font></i></a></i>;<br/>
<a NAME="E81:38_1"/><i><font color="Green" title="E80">A53</font></i>: 
<font color="Maroon" title="c24">L3a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">L3b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1_2"/>
<font color="Maroon" title="c24">L3a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">L3b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:38_1_2"/><b>then </b>
<font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">L3b</font>
 
<b>by </b><i><a class="txt" href="#E77:38_1"><i><font color="Green" title="E78">A51</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E3:38_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">A2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c19">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">L3b</font>
 
<b>by </b><i><a class="txt" href="#E77:38_1"><i><font color="Green" title="E78">A51</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E4:38_1_2"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="#E77:38_1"><i><font color="Green" title="E78">A51</font></i></a>, <a class="txt" href="#E80:38_1"><i><font color="Green" title="E79">A52</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E5:38_1_2"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/>
<b>hence </b><a NAME="E6:38_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E82:38_1"/>
( <font color="Maroon" title="c17">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">L1</font> &amp; <font color="Maroon" title="c18">A2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">L1</font> &amp; <font color="Maroon" title="c19">A3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">L1</font> )
 
<b>by </b><i><a class="txt" href="#E15:38_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="#E27:38_1"><i><font color="Green" title="E53">A26</font></i></a>, <a class="txt" href="#E44:38_1"><i><font color="Green" title="E62">A35</font></i></a>, <a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">Def8</a></i>;<br/>
<a NAME="E83:38_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c17">A1</font>,<font color="Maroon" title="c18">A2</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">L1</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">A2</font>,<font color="Maroon" title="c19">A3</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">L1</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></i>;<br/>
<a NAME="E84:38_1"/><b>then </b>
( <font color="Maroon" title="c20">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">L3a</font> &amp; <font color="Maroon" title="c20">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">L3b</font> )
 
<b>by </b><i><a class="txt" href="#E18:38_1"><i><font color="Green" title="E48">A21</font></i></a>, <a class="txt" href="#E20:38_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="#E22:38_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="txt" href="#E24:38_1"><i><font color="Green" title="E51">A24</font></i></a>, <a class="txt" href="#E29:38_1"><i><font color="Green" title="E54">A27</font></i></a>, <a class="txt" href="#E31:38_1"><i><font color="Green" title="E55">A28</font></i></a>, <a class="txt" href="#E33:38_1"><i><font color="Green" title="E56">A29</font></i></a>, <a class="txt" href="#E37:38_1"><i><font color="Green" title="E58">A31</font></i></a>, <a class="txt" href="#E39:38_1"><i><font color="Green" title="E59">A32</font></i></a>, <a class="txt" href="#E41:38_1"><i><font color="Green" title="E60">A33</font></i></a>, <a class="txt" href="#E55:38_1"><i><font color="Green" title="E68">A41</font></i></a>, <a class="txt" href="#E57:38_1"><i><font color="Green" title="E69">A42</font></i></a>, <a class="ref" href="incsp_1.html#D10" title="INCSP_1:def.10">INCSP_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E85:38_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E81:38_1"><i><font color="Green" title="E80">A53</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E13:38"/>
not <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font> is <a href="combgras.html#V4" title="COMBGRAS:attr.4">TOP</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_2"/>
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font> is <a href="combgras.html#V4" title="COMBGRAS:attr.4">TOP</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c5">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E3:38_2"/><i><font color="Green" title="E39">A54</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> : (  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="combgras.html#D5" target="_self" title="COMBGRAS:def.5">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:38_2"/><i><font color="Green" title="E40">A55</font></i>: 
( <font color="Maroon" title="c6">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="#E6:38"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="#E3:38_2"><i><font color="Green" title="E39">A54</font></i></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></i>;<br/>
<a NAME="E6:38_2"/><i><font color="Green" title="E41">A56</font></i>: 
( <font color="Maroon" title="c6">X1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c5">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="txt" href="#E3:38_2"><i><font color="Green" title="E39">A54</font></i></a>, <a class="txt" href="#E5:38_2"><i><font color="Green" title="E40">A55</font></i></a></i>;<br/>
<a NAME="E7:38_2"/><b>then </b><i><font color="Green" title="E42">A57</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</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> 1
 
<b>by </b><i><a class="txt" href="#E3:38_2"><i><font color="Green" title="E39">A54</font></i></a>, <a class="txt" href="#E5:38_2"><i><font color="Green" title="E40">A55</font></i></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">X2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:38_2"/><i><font color="Green" title="E43">A58</font></i>: 
( <font color="Maroon" title="c7">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="#E6:38"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></i>;<br/>
<a NAME="E10:38_2"/>
( <font color="Maroon" title="c7">X2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="txt" href="#E7:38_2"><i><font color="Green" title="E42">A57</font></i></a>, <a class="txt" href="#E9:38_2"><i><font color="Green" title="E43">A58</font></i></a></i>;<br/>
<a NAME="E11:38_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="txt" href="#E7:38_2"><i><font color="Green" title="E42">A57</font></i></a>, <a class="txt" href="#E9:38_2"><i><font color="Green" title="E43">A58</font></i></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">X3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:38_2"/><i><font color="Green" title="E44">A60</font></i>: 
( <font color="Maroon" title="c8">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c8">X3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="#E6:38"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></i>;<br/>
<a NAME="E14:38_2"/>
( <font color="Maroon" title="c8">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">X2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">X1</font></span>)</span> &amp; the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> &amp; the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> )
 
<b>by </b><i><a class="txt" href="#E2:38"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="#E13:38_2"><i><font color="Green" title="E44">A60</font></i></a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a>, <a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a></i>;<br/>
<a NAME="E15:38_2"/><b>then </b><i><font color="Green" title="E45">A61</font></i>: 
( <font color="Maroon" title="c8">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c7">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c8">X3</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c8">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> &amp; the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; <font color="Maroon" title="c3">F</font> is <a href="combgras.html#V6" title="COMBGRAS:attr.6">incidence_preserving</a> &amp; the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> is <a href="funct_1.html#V1" title="FUNCT_1:attr.1">Function-like</a> )
 
<b>by </b><i><a class="txt" href="#E2:38"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="#E9:38_2"><i><font color="Green" title="E43">A58</font></i></a>, <a class="txt" href="#E13:38_2"><i><font color="Green" title="E44">A60</font></i></a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a>, <a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a>, <a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></i>;<br/>
<a NAME="E16:38_2"/><b>then </b><i><font color="Green" title="E46">A62</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</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> 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="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</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> 1 &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c8">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> &amp; <font color="Maroon" title="c8">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> &amp; <font color="Maroon" title="c6">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">X2</font> &amp; <font color="Maroon" title="c7">X2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">X3</font> &amp; <font color="Maroon" title="c6">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">X3</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> &amp; 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>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; 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>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> &amp; 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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; 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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> )
 
<b>by </b><i><a class="txt" href="#E3:38_2"><i><font color="Green" title="E39">A54</font></i></a>, <a class="txt" href="#E6:38_2"><i><font color="Green" title="E41">A56</font></i></a>, <a class="txt" href="#E9:38_2"><i><font color="Green" title="E43">A58</font></i></a>, <a class="txt" href="#E13:38_2"><i><font color="Green" title="E44">A60</font></i></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a>, <a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a>, <a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></i>;<br/>
<a NAME="E17:38_2"/><b>then </b><i><font color="Green" title="E47">A63</font></i>: 
( <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</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> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</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> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</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> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E7:38"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="#E8:38"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="#E3:38_2"><i><font color="Green" title="E39">A54</font></i></a>, <a class="txt" href="#E7:38_2"><i><font color="Green" title="E42">A57</font></i></a></i>;<br/>
<a NAME="E18:38_2"/><i><font color="Green" title="E48">A64</font></i>: 
( <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_2_1"/>
( <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> or <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</font> or <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:38_2_1"/><b>then </b>
( <font color="Maroon" title="c7">X2</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">X3</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">X3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="#E9:38_2"><i><font color="Green" title="E43">A58</font></i></a>, <a class="txt" href="#E16:38_2"><i><font color="Green" title="E46">A62</font></i></a>, <a class="ref" href="xboole_1.html#T38" title="XBOOLE_1:th.38">XBOOLE_1:38</a></i>;<br/>
<b>hence </b><a NAME="E3:38_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E9:38_2"><i><font color="Green" title="E43">A58</font></i></a>, <a class="txt" href="#E13:38_2"><i><font color="Green" title="E44">A60</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c9">B1</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E20:38_2"/><i><font color="Green" title="E49">A65</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B1</font>
 <b>by </b><i><a class="txt" href="#E17:38_2"><i><font color="Green" title="E47">A63</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">B2</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E22:38_2"/><i><font color="Green" title="E50">A66</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B2</font>
 <b>by </b><i><a class="txt" href="#E17:38_2"><i><font color="Green" title="E47">A63</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">B3</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E24:38_2"/><i><font color="Green" title="E51">A67</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">X3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">B3</font>
 <b>by </b><i><a class="txt" href="#E17:38_2"><i><font color="Green" title="E47">A63</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">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><b> such that </b><br/><a NAME="E26:38_2"/><i><font color="Green" title="E52">A68</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">L2</font>
 <b>by </b><i><a class="txt" href="#E17:38_2"><i><font color="Green" title="E47">A63</font></i></a></i>;<br/>
<a NAME="E27:38_2"/><i><font color="Green" title="E53">A69</font></i>: 
( <font color="Maroon" title="c9">B1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">L2</font> &amp; <font color="Maroon" title="c10">B2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">L2</font> &amp; <font color="Maroon" title="c11">B3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">L2</font> &amp; <font color="Maroon" title="c9">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c10">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c11">B3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c10">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Maroon" title="c12">L2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c9">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> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">B3</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> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c12">L2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E5:38"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E16:38_2"><i><font color="Green" title="E46">A62</font></i></a>, <a class="txt" href="#E20:38_2"><i><font color="Green" title="E49">A65</font></i></a>, <a class="txt" href="#E22:38_2"><i><font color="Green" title="E50">A66</font></i></a>, <a class="txt" href="#E24:38_2"><i><font color="Green" title="E51">A67</font></i></a>, <a class="txt" href="#E26:38_2"><i><font color="Green" title="E52">A68</font></i></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">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><b> such that </b><br/><a NAME="E29:38_2"/><i><font color="Green" title="E54">A70</font></i>: 
<font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c9">B1</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c14">A2</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E31:38_2"/><i><font color="Green" title="E55">A71</font></i>: 
<font color="Maroon" title="c14">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c10">B2</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c15">A3</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E33:38_2"/><i><font color="Green" title="E56">A72</font></i>: 
<font color="Maroon" title="c15">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">B3</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c16">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><b> such that </b><br/><a NAME="E35:38_2"/><i><font color="Green" title="E57">A73</font></i>: 
<font color="Maroon" title="c16">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c12">L2</font>
 ;<br/>
<a NAME="E36:38_2"/><i><font color="Green" title="E58">A74</font></i>: 
( <font color="Maroon" title="c13">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">L1</font> &amp; <font color="Maroon" title="c14">A2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">L1</font> &amp; <font color="Maroon" title="c15">A3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">L1</font> )
 
<b>by </b><i><a class="txt" href="#E15:38_2"><i><font color="Green" title="E45">A61</font></i></a>, <a class="txt" href="#E27:38_2"><i><font color="Green" title="E53">A69</font></i></a>, <a class="txt" href="#E29:38_2"><i><font color="Green" title="E54">A70</font></i></a>, <a class="txt" href="#E31:38_2"><i><font color="Green" title="E55">A71</font></i></a>, <a class="txt" href="#E33:38_2"><i><font color="Green" title="E56">A72</font></i></a>, <a class="txt" href="#E35:38_2"><i><font color="Green" title="E57">A73</font></i></a>, <a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">Def8</a></i>;<br/>
<a NAME="E37:38_2"/><b>then </b><i><font color="Green" title="E59">A75</font></i>: 
( <font color="Maroon" title="c13">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">L1</font> &amp; <font color="Maroon" title="c14">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">L1</font> &amp; <font color="Maroon" title="c15">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">L1</font> &amp; <font color="Maroon" title="c9">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c13">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">A2</font> &amp; <font color="Maroon" title="c14">A2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">A3</font> &amp; <font color="Maroon" title="c13">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">A3</font> &amp; <font color="Maroon" title="c10">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c11">B3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font> )
 
<b>by </b><i><a class="txt" href="#E5:38"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E3:38_2"><i><font color="Green" title="E39">A54</font></i></a>, <a class="txt" href="#E7:38_2"><i><font color="Green" title="E42">A57</font></i></a>, <a class="txt" href="#E15:38_2"><i><font color="Green" title="E45">A61</font></i></a>, <a class="txt" href="#E16:38_2"><i><font color="Green" title="E46">A62</font></i></a>, <a class="txt" href="#E18:38_2"><i><font color="Green" title="E48">A64</font></i></a>, <a class="txt" href="#E20:38_2"><i><font color="Green" title="E49">A65</font></i></a>, <a class="txt" href="#E22:38_2"><i><font color="Green" title="E50">A66</font></i></a>, <a class="txt" href="#E24:38_2"><i><font color="Green" title="E51">A67</font></i></a>, <a class="txt" href="#E29:38_2"><i><font color="Green" title="E54">A70</font></i></a>, <a class="txt" href="#E31:38_2"><i><font color="Green" title="E55">A71</font></i></a>, <a class="txt" href="#E33:38_2"><i><font color="Green" title="E56">A72</font></i></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E38:38_2"/><b>then </b><i><font color="Green" title="E60">A76</font></i>: 
( <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c14">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c15">A3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> )
 
<b>by </b><i><a class="txt" href="#E29:38_2"><i><font color="Green" title="E54">A70</font></i></a>, <a class="txt" href="#E31:38_2"><i><font color="Green" title="E55">A71</font></i></a>, <a class="txt" href="#E33:38_2"><i><font color="Green" title="E56">A72</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">L3a</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E40:38_2"/><i><font color="Green" title="E61">A77</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">A1</font>,<font color="Maroon" title="c14">A2</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">L3a</font>
 <b>by </b><i><a class="txt" href="#E11:38"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">L3b</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E42:38_2"/><i><font color="Green" title="E62">A78</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">A2</font>,<font color="Maroon" title="c15">A3</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">L3b</font>
 <b>by </b><i><a class="txt" href="#E11:38"><i><font color="Green" title="E37">A9</font></i></a>, <a class="txt" href="#E38:38_2"><i><font color="Green" title="E60">A76</font></i></a>, <a class="ref" href="combgras.html#D2" target="_self" title="COMBGRAS:def.2">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">S</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E44:38_2"/><i><font color="Green" title="E63">A79</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c19">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; <font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> : (  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c19">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="#E9:38"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="combgras.html#D4" target="_self" title="COMBGRAS:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">A11</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E46:38_2"/><i><font color="Green" title="E64">A80</font></i>: 
( <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">A11</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c20">A11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c19">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">A11</font> )
 <b>by </b><i><a class="txt" href="#E38:38_2"><i><font color="Green" title="E60">A76</font></i></a>, <a class="txt" href="#E44:38_2"><i><font color="Green" title="E63">A79</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">A12</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E48:38_2"/><i><font color="Green" title="E65">A81</font></i>: 
( <font color="Maroon" title="c14">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">A12</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c21">A12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c19">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">A12</font> )
 <b>by </b><i><a class="txt" href="#E38:38_2"><i><font color="Green" title="E60">A76</font></i></a>, <a class="txt" href="#E44:38_2"><i><font color="Green" title="E63">A79</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">A13</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E50:38_2"/><i><font color="Green" title="E66">A82</font></i>: 
( <font color="Maroon" title="c15">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">A13</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c22">A13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c19">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">A13</font> )
 <b>by </b><i><a class="txt" href="#E38:38_2"><i><font color="Green" title="E60">A76</font></i></a>, <a class="txt" href="#E44:38_2"><i><font color="Green" title="E63">A79</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c23">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E52:38_2"/><i><font color="Green" title="E67">A83</font></i>: 
<font color="Maroon" title="c23">m</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
 <b>by </b><i><a class="txt" href="#E5:38"><i><font color="Green" title="E32">A4</font></i></a></i>;<br/>
<a NAME="E53:38_2"/>
<font color="Maroon" title="c16">L1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c24">l12</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E55:38_2"/><i><font color="Green" title="E68">A84</font></i>: 
( <font color="Maroon" title="c16">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">l12</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c24">l12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="#E8:38"><i><font color="Green" title="E35">A7</font></i></a></i>;<br/>
<a NAME="E56:38_2"/>
( <font color="Maroon" title="c19">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">A2</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">A2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">L1</font> &amp; <font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">L1</font> &amp; <font color="Maroon" title="c14">A2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">L1</font> &amp; 2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> )
 
<b>by </b><i><a class="txt" href="#E1:38"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E37:38_2"><i><font color="Green" title="E59">A75</font></i></a>, <a class="txt" href="#E46:38_2"><i><font color="Green" title="E64">A80</font></i></a>, <a class="txt" href="#E48:38_2"><i><font color="Green" title="E65">A81</font></i></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a>, <a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E57:38_2"/><b>then </b>
( <font color="Maroon" title="c19">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c15">A3</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">A2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c15">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">A2</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">A2</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 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">A3</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 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">A2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">A3</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 &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">A2</font></span>)</span> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">A3</font></span>)</span> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">A2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">A3</font></span>)</span> &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">k</font> )
 
<b>by </b><i><a class="txt" href="#E6:38"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="#E37:38_2"><i><font color="Green" title="E59">A75</font></i></a>, <a class="txt" href="#E46:38_2"><i><font color="Green" title="E64">A80</font></i></a>, <a class="txt" href="#E48:38_2"><i><font color="Green" title="E65">A81</font></i></a>, <a class="txt" href="#E50:38_2"><i><font color="Green" title="E66">A82</font></i></a>, <a class="txt" href="#E55:38_2"><i><font color="Green" title="E68">A84</font></i></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</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#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E58:38_2"/><b>then </b><i><font color="Green" title="E69">A85</font></i>: 
( <font color="Maroon" title="c23">m</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c15">A3</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">A2</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c23">m</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">A2</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 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">A2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">A3</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 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">A3</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 &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="c13">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c15">A3</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">A2</font></span>)</span> &amp; ( 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> or <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 
<b>by </b><i><a class="txt" href="#E44:38_2"><i><font color="Green" title="E63">A79</font></i></a>, <a class="txt" href="#E52:38_2"><i><font color="Green" title="E67">A83</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="ordinal1.html#T13" title="ORDINAL1:th.13">ORDINAL1:13</a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E59:38_2"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c15">A3</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c23">m</font> &amp; ( 3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> or <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 
<b>by </b><i><a class="txt" href="#E6:38"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E60:38_2"/><b>then </b><i><font color="Green" title="E70">A86</font></i>: 
( <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c13">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c15">A3</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c13">A1</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="c14">A2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">A3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">A3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c14">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c15">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )
 
<b>by </b><i><a class="txt" href="#E46:38_2"><i><font color="Green" title="E64">A80</font></i></a>, <a class="txt" href="#E48:38_2"><i><font color="Green" title="E65">A81</font></i></a>, <a class="txt" href="#E50:38_2"><i><font color="Green" title="E66">A82</font></i></a>, <a class="txt" href="#E52:38_2"><i><font color="Green" title="E67">A83</font></i></a>, <a class="txt" href="#E58:38_2"><i><font color="Green" title="E69">A85</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E61:38_2"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">A2</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> <font color="Maroon" title="c13">A1</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;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">A2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c15">A3</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"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c15">A3</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> <font color="Maroon" title="c14">A2</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;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c15">A3</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 &amp; 2 <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 &amp; <font color="Maroon" title="c1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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; <font color="Maroon" title="c13">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">L3a</font> &amp; <font color="Maroon" title="c14">A2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">L3b</font> &amp; <font color="Maroon" title="c15">A3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">L3b</font> )
 
<b>by </b><i><a class="txt" href="#E1:38"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E3:38"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="#E5:38"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E40:38_2"><i><font color="Green" title="E61">A77</font></i></a>, <a class="txt" href="#E42:38_2"><i><font color="Green" title="E62">A78</font></i></a>, <a class="ref" href="combgras.html#T2" target="_self" title="COMBGRAS:th.2">Th2</a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E62:38_2"/><b>then </b><i><font color="Green" title="E71">A87</font></i>: 
(  <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="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">A2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">A3</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 &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="c13">A1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">A2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c15">A3</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; <font color="Maroon" title="c13">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">L3a</font> &amp; <font color="Maroon" title="c14">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">L3b</font> &amp; <font color="Maroon" title="c15">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">L3b</font> )
 
<b>by </b><i><a class="txt" href="#E5:38"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E60:38_2"><i><font color="Green" title="E70">A86</font></i></a>, <a class="ref" href="combgras.html#T7" target="_self" title="COMBGRAS:th.7">Th7</a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></i>;<br/>
<a NAME="E63:38_2"/>
<font color="Maroon" title="c18">L3b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c25">l13b</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E65:38_2"/><i><font color="Green" title="E72">A88</font></i>: 
( <font color="Maroon" title="c18">L3b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">l13b</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c25">l13b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="#E8:38"><i><font color="Green" title="E35">A7</font></i></a></i>;<br/>
<a NAME="E66:38_2"/><i><font color="Green" title="E73">A89</font></i>: 
<font color="Maroon" title="c17">L3a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">L3b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_2_2"/>
<font color="Maroon" title="c17">L3a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">L3b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:38_2_2"/><b>then </b>
<font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">L3b</font>
 
<b>by </b><i><a class="txt" href="#E62:38_2"><i><font color="Green" title="E71">A87</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E3:38_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">A2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">L3b</font>
 
<b>by </b><i><a class="txt" href="#E62:38_2"><i><font color="Green" title="E71">A87</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E4:38_2_2"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="#E62:38_2"><i><font color="Green" title="E71">A87</font></i></a>, <a class="txt" href="#E65:38_2"><i><font color="Green" title="E72">A88</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E5:38_2_2"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/>
<b>hence </b><a NAME="E6:38_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E67:38_2"/>
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">A1</font>,<font color="Maroon" title="c14">A2</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">L1</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">A2</font>,<font color="Maroon" title="c15">A3</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">L1</font> )
 
<b>by </b><i><a class="txt" href="#E36:38_2"><i><font color="Green" title="E58">A74</font></i></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></i>;<br/>
<a NAME="E68:38_2"/><b>then </b>
( <font color="Maroon" title="c16">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">L3a</font> &amp; <font color="Maroon" title="c16">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">L3b</font> )
 
<b>by </b><i><a class="txt" href="#E37:38_2"><i><font color="Green" title="E59">A75</font></i></a>, <a class="txt" href="#E40:38_2"><i><font color="Green" title="E61">A77</font></i></a>, <a class="txt" href="#E42:38_2"><i><font color="Green" title="E62">A78</font></i></a>, <a class="ref" href="incsp_1.html#D10" title="INCSP_1:def.10">INCSP_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E69:38_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E66:38_2"><i><font color="Green" title="E73">A89</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:38"/>
( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">K</font> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a> )
 <b>by </b><i><a class="txt" href="#E1:38"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E11:38"><i><font color="Green" title="E37">A9</font></i></a>, <a class="txt" href="#E12:38"><i><font color="Green" title="E38">A10</font></i></a>, <a class="ref" href="combgras.html#T15" target="_self" title="COMBGRAS:th.15">Th15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
