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

<span class="kw">let </span><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="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>) holds <br/> ( <font color="Olive" title="b2">F</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a>  iff  ex <font color="Olive" title="b3">s</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Olive" title="b1">X</font> st  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b2">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b2">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b3">s</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> 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>) holds <br/> ( <font color="Olive" title="b1">F</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a>  iff  ex <font color="Olive" title="b2">s</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font> st  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b1">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b1">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b2">s</font>) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:51"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">and </span><br/><a NAME="E2:51"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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>) holds <br/> ( <font color="Olive" title="b1">F</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a>  iff  ex <font color="Olive" title="b2">s</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font> st  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b1">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b1">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b2">s</font>) )</span><br/>

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a>  iff  ex <font color="Olive" title="b1">s</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font> st  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">s</font>) )</span><br/>

<a NAME="E3:51"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
( <font color="Maroon" title="c3">F</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a>  implies  ex <font color="Olive" title="b1">s</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font> st  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">s</font>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:51_1"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
(  <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> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D2" title="CARD_1:def.2">CARD_1:def 2</a></span>;<br/>
<a NAME="E2:51_1"/><span class="lab"><font color="Green" title="E44">A5</font></span>: 
 <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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D2" title="CARD_1:def.2">CARD_1:def 2</a></span>;<br/>
<a NAME="E3:51_1"/>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>;<br/>
<a NAME="E4:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A6</font></span>: 
( <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font> or <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T8" title="ORDINAL1:th.8">ORDINAL1:8</a></span>;<br/>
<a NAME="E5:51_1"/><span class="lab"><font color="Green" title="E46">A7</font></span>: 
 <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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D2" title="CARD_1:def.2">CARD_1:def 2</a></span>;<br/>
<a NAME="E6:51_1"/>
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="nat_1.html#T38" title="NAT_1:th.38">NAT_1:38</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E7:51_1"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="nat_1.html#T41" title="NAT_1:th.41">NAT_1:41</a></span>;<br/>
<a NAME="E8:51_1"/><span class="kw">then </span>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> or 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T8" title="ORDINAL1:th.8">ORDINAL1:8</a></span>;<br/>
<a NAME="E9:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A8</font></span>: 
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> or ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> &amp; 2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E1:51_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="ref" href="nat_1.html#T41" title="NAT_1:th.41">NAT_1:41</a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/>
<span class="kw">assume </span><a NAME="E10:51_1"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font> st  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">s</font>)</span><br/>

<a NAME="E11:51_1"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T38" title="NAT_1:th.38">NAT_1:38</a></span>;<br/>
<a NAME="E12:51_1"/><span class="kw">then </span>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> or ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</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 ) or ( 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> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="txt" href="#E9:51_1"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:51_1"/>
 ex <font color="Olive" title="b1">s</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font> st  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">s</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E10:51_1"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="ref" href="combgras.html#T24" target="_self" title="COMBGRAS:th.24">Th24</a>, <a class="ref" href="combgras.html#T25" target="_self" title="COMBGRAS:th.25">Th25</a>, <a class="ref" href="combgras.html#T33" target="_self" title="COMBGRAS:th.33">Th33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:51"/>
(  ex <font color="Olive" title="b1">s</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font> st  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">s</font>) implies <font color="Maroon" title="c3">F</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:51_2"/>
 ex <font color="Olive" title="b1">s</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font> st  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">s</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">F</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">s</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font><span class="kw"> such that </span><br/><a NAME="E3:51_2"/><span class="lab"><font color="Green" title="E43">A10</font></span>: 
 <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">s</font>)
 ;<br/>
<a NAME="E4:51_2"/><span class="lab"><font color="Green" title="E44">A11</font></span>: 
 <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">s</font>) is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="combgras.html#T34" target="_self" title="COMBGRAS:th.34">Th34</a></span>;<br/>
<a NAME="E5:51_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A12</font></span>: 
 <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">s</font>) is  <a href="combgras.html#V6" title="COMBGRAS:attr.6">incidence_preserving</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a></span>;<br/>
<a NAME="E6:51_2"/><span class="lab"><font color="Green" title="E46">A13</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="combgras.html#V6" title="COMBGRAS:attr.6">incidence_preserving</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">A</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font>)</span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">COMBGRAS:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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> holds <br/> ( <font color="Maroon" title="c5">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">L1</font> iff <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c5">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Olive" title="b1">L1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">L</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font> iff <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c5">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c6">L</font> )</span><br/>



<a NAME="E1:51_2_1"/>
( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c6">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:51_2_1"/>
( <font color="Maroon" title="c5">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font> iff <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c5">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c6">L</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_2"><span class="lab"><font color="Green" title="E45">A12</font></span></a>, <a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:51_2"/>
(  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>  &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>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E4:51_2"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:51_2"/>
<font color="Maroon" title="c3">F</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_2"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:51"/>
( <font color="Maroon" title="c3">F</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a>  iff  ex <font color="Olive" title="b1">s</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font> st  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(#  the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font>, the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> (<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">s</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E42">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
