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

<b>let </b><font color="Maroon" title="c1">S</font> be    <a href="incsp_1.html#L1" title="INCSP_1:struct.1">IncProjStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">F</font> be    <a href="combgras.html#L1" title="COMBGRAS:struct.1">IncProjMap</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">K</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">S</font>;<br/>





<b>assume </b><a NAME="E1:37"/><i><font color="Green" title="E28">A1</font></i>: 
( <font color="Maroon" title="c2">F</font> is <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> &amp; <font color="Maroon" title="c3">K</font> is <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a> )
 ;<br/>

<a NAME="E2:37"/><b>then </b>
( <font color="Maroon" title="c2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> :  ex <font color="Olive" title="b2">A</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c2">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> ) </span>}</span>  &amp; the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c2">F</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> &amp; the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c2">F</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> )
 
<b>by </b><i><a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a>, <a class="ref" href="combgras.html#T18" target="_self" title="COMBGRAS:th.18">Th18</a></i>;<br/>
<a NAME="E3:37"/><b>then </b><i><font color="Green" title="E29">A2</font></i>: 
( <font color="Maroon" title="c3">K</font> is <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="combgras.html#V6" title="COMBGRAS:attr.6">incidence_preserving</a> &amp; the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c2">F</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c2">F</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 
<b>by </b><i><a class="txt" href="combgras.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="combgras.html#D3" target="_self" title="COMBGRAS:def.3">Def3</a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E4:37"/><b>then </b><i><font color="Green" title="E30">A3</font></i>: 
( <font color="Maroon" title="c2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c3">K</font> is <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> &amp; <font color="Maroon" title="c2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c3">K</font> is <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> &amp; the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">S</font> <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="c2">F</font> &amp; the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <font color="Maroon" title="c1">S</font> <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="c2">F</font> &amp; the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <font color="Maroon" title="c1">S</font> <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="c2">F</font> &amp; the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">S</font> <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="c2">F</font> )
 
<b>by </b><i><a class="ref" href="combgras.html#T20" target="_self" title="COMBGRAS:th.20">Th20</a>, <a class="ref" href="combgras.html#T21" target="_self" title="COMBGRAS:th.21">Th21</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></i>;<br/>
<a NAME="E5:37"/><i><font color="Green" title="E31">A4</font></i>: 
for <font color="Olive" title="b1">U</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">U</font> is <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> &amp; <font color="Maroon" title="c2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U</font> holds <br/><font color="Olive" title="b1">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c3">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">U</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">S</font>;<br/>

<b>assume </b><a NAME="E1:37_1"/><i><font color="Green" title="E32">A5</font></i>: 
( <font color="Maroon" title="c4">U</font> is <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> &amp; <font color="Maroon" title="c2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> )
 ;<br/>

<a NAME="E2:37_1"/><i><font color="Green" title="E33">A6</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">U</font> is <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a>
 
<b>by </b><i><a class="txt" href="combgras.html#E3:37"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="combgras.html#E1:37_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="combgras.html#T21" target="_self" title="COMBGRAS:th.21">Th21</a></i>;<br/>
<a NAME="E3:37_1"/><i><font color="Green" title="E34">A7</font></i>: 
( <font color="Maroon" title="c2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">U</font> &amp; <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">F</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E4:37"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="combgras.html#E1:37_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="relat_1.html#T178" title="RELAT_1:th.178">RELAT_1:178</a></i>;<br/>
<a NAME="E4:37_1"/>
<font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c3">K</font></span>)</span>
 
<b>by </b><i><a class="txt" href="combgras.html#E4:37"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="funct_1.html#T146" title="FUNCT_1:th.146">FUNCT_1:146</a></i>;<br/>
<a NAME="E5:37_1"/><b>then </b>
<font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">U</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E3:37_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:37_1"/><b>then </b>
( <font color="Maroon" title="c2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c4">U</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c4">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">F</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="combgras.html#E4:37"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="combgras.html#E2:37_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="combgras.html#D3" target="_self" title="COMBGRAS:def.3">Def3</a></i>;<br/>
<a NAME="E7:37_1"/><b>then </b>
<font color="Maroon" title="c4">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c3">K</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T147" title="FUNCT_1:th.147">FUNCT_1:147</a></i>;<br/>
<b>hence </b><a NAME="E8:37_1"/>
<font color="Maroon" title="c4">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c3">K</font>
 <b>by </b><i><a class="txt" href="combgras.html#E1:37_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:37"/>
for <font color="Olive" title="b1">U</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">U</font> is <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> &amp; <font color="Maroon" title="c2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U</font> holds <br/><font color="Olive" title="b1">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c3">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">U</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">S</font>;<br/>

<b>assume </b><a NAME="E1:37_2"/><i><font color="Green" title="E32">A8</font></i>: 
( <font color="Maroon" title="c4">U</font> is <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a> &amp; <font color="Maroon" title="c2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> )
 ;<br/>

<a NAME="E2:37_2"/><i><font color="Green" title="E33">A9</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">U</font> is <a href="combgras.html#V1" title="COMBGRAS:attr.1">clique</a>
 
<b>by </b><i><a class="txt" href="combgras.html#E3:37"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="combgras.html#E1:37_2"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="combgras.html#T20" target="_self" title="COMBGRAS:th.20">Th20</a></i>;<br/>
<a NAME="E3:37_2"/>
( <font color="Maroon" title="c2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">U</font> &amp; <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">F</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E4:37"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="combgras.html#E1:37_2"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<a NAME="E4:37_2"/><b>then </b>
<font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">U</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T147" title="FUNCT_1:th.147">FUNCT_1:147</a></i>;<br/>
<a NAME="E5:37_2"/><b>then </b>
( <font color="Maroon" title="c2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">U</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c4">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c4">U</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="combgras.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="combgras.html#E4:37"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="combgras.html#E2:37_2"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="combgras.html#D3" target="_self" title="COMBGRAS:def.3">Def3</a>, <a class="ref" href="funct_1.html#T146" title="FUNCT_1:th.146">FUNCT_1:146</a></i>;<br/>
<a NAME="E6:37_2"/><b>then </b>
<font color="Maroon" title="c4">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c3">K</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E7:37_2"/>
<font color="Maroon" title="c4">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c3">K</font>
 <b>by </b><i><a class="txt" href="combgras.html#E1:37_2"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:37"/>
( <font color="Maroon" title="c2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <font color="Maroon" title="c3">K</font> is <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a> &amp; <font color="Maroon" title="c2">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <font color="Maroon" title="c3">K</font> is <a href="combgras.html#V2" title="COMBGRAS:attr.2">maximal_clique</a> )
 <b>by </b><i><a class="txt" href="combgras.html#E4:37"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="combgras.html#E5:37"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="combgras.html#D3" target="_self" title="COMBGRAS:def.3">Def3</a></i>;<br/>


</div>
