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

<b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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  <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> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <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> <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>,<font color="Olive" title="b1">X</font>, <a href="combgras.html#K1" title="COMBGRAS:func.1">G_</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>,<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">H</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="b3">H</font> is <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> &amp; the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b3">H</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b2">F</font> &amp; ( for <font color="Olive" title="b4">A</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">X</font></span>)</span><br/> for <font color="Olive" title="b5">B</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">A</font> holds <br/><font color="Olive" title="b3">H</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Olive" title="b4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p2">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Olive" title="b5">B</font>,<font color="Olive" title="b1">X</font>,<span class="p3">(<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></span>)</span></span>)</span> ) holds <br/>for <font color="Olive" title="b4">f</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="b3">H</font>,the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b3">H</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="b4">f</font> holds <br/> <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> <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>,<font color="Olive" title="b4">f</font></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"> (  <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> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <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> <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>,<font color="Maroon" title="c2">X</font>, <a href="combgras.html#K1" title="COMBGRAS:func.1">G_</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>,<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">H</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="b2">H</font> is <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> &amp; the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b2">H</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b1">F</font> &amp; ( for <font color="Olive" title="b3">A</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><br/> for <font color="Olive" title="b4">B</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A</font> holds <br/><font color="Olive" title="b2">H</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p2">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Olive" title="b4">B</font>,<font color="Maroon" title="c2">X</font>,<span class="p3">(<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></span>)</span></span>)</span> ) holds <br/>for <font color="Olive" title="b3">f</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="b2">H</font>,the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b2">H</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">f</font> holds <br/> <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> <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>,<font color="Olive" title="b3">f</font> )</span><br/>



<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E38">A1</font></i>: 
(  <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> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <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> <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>,<font color="Maroon" title="c2">X</font>, <a href="combgras.html#K1" title="COMBGRAS:func.1">G_</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>,<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">H</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="b2">H</font> is <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> &amp; the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b2">H</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b1">F</font> &amp; ( for <font color="Olive" title="b3">A</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><br/> for <font color="Olive" title="b4">B</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A</font> holds <br/><font color="Olive" title="b2">H</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p2">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Olive" title="b4">B</font>,<font color="Maroon" title="c2">X</font>,<span class="p3">(<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></span>)</span></span>)</span> ) holds <br/>for <font color="Olive" title="b3">f</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="b2">H</font>,the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b2">H</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">f</font> holds <br/> <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> <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>,<font color="Olive" title="b3">f</font></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> <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>,<font color="Maroon" title="c2">X</font>, <a href="combgras.html#K1" title="COMBGRAS:func.1">G_</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>,<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">H</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">H</font> is <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> &amp; the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b1">H</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; ( for <font color="Olive" title="b2">A</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><br/> for <font color="Olive" title="b3">B</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> holds <br/><font color="Olive" title="b1">H</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>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p2">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Olive" title="b3">B</font>,<font color="Maroon" title="c2">X</font>,<span class="p3">(<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></span>)</span></span>)</span> ) holds <br/>for <font color="Olive" title="b2">f</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">H</font>,the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b1">H</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">f</font> holds <br/> <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> <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>,<font color="Olive" title="b2">f</font> )</span><br/>

<b>assume </b><a NAME="E2:48"/><i><font color="Green" title="E39">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">H</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">H</font> is <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> &amp; the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b1">H</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; ( for <font color="Olive" title="b2">A</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><br/> for <font color="Olive" title="b3">B</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> holds <br/><font color="Olive" title="b1">H</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>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p2">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Olive" title="b3">B</font>,<font color="Maroon" title="c2">X</font>,<span class="p3">(<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></span>)</span></span>)</span> ) holds <br/>for <font color="Olive" title="b2">f</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">H</font>,the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b1">H</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">f</font> holds <br/> <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> <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>,<font color="Olive" title="b2">f</font></span><br/>

<b>let </b><font color="Maroon" title="c4">H</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="c4">H</font> is <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> &amp; the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c4">H</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><br/> for <font color="Olive" title="b2">B</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Maroon" title="c4">H</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p2">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Olive" title="b2">B</font>,<font color="Maroon" title="c2">X</font>,<span class="p3">(<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></span>)</span></span>)</span> ) implies for <font color="Olive" title="b1">f</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="c4">H</font>,the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c4">H</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">f</font> holds <br/> <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> <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>,<font color="Olive" title="b1">f</font> )</span><br/>

<b>assume </b><a NAME="E3:48"/><i><font color="Green" title="E40">A3</font></i>: 
( <font color="Maroon" title="c4">H</font> is <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> &amp; the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c4">H</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><br/> for <font color="Olive" title="b2">B</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Maroon" title="c4">H</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p2">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Olive" title="b2">B</font>,<font color="Maroon" title="c2">X</font>,<span class="p3">(<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></span>)</span></span>)</span> ) )
 ; <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="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="c4">H</font>,the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c4">H</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">f</font> holds <br/> <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> <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>,<font color="Olive" title="b1">f</font></span><br/>

<b>let </b><font color="Maroon" title="c5">f</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <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"> (  <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="c4">H</font>,the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c4">H</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="c5">f</font> implies  <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> <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>,<font color="Maroon" title="c5">f</font> )</span><br/>

<b>assume </b><a NAME="E4:48"/><i><font color="Green" title="E41">A4</font></i>: 
 <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="c4">H</font>,the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c4">H</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="c5">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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> <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>,<font color="Maroon" title="c5">f</font></span><br/>

<a NAME="E5:48"/>
( <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> 3 &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <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> 3 &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="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E6:48"/><b>then </b><i><font color="Green" title="E42">A5</font></i>: 
( <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 &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> 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> 3 &amp; <font color="Maroon" title="c1">k</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</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> 2 )
 
<b>by </b><i><a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a>, <a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/>
<a NAME="E7:48"/><b>then </b><i><font color="Green" title="E43">A6</font></i>: 
( <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; <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 <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;  <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="txt" href="combgras.html#E1:48"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:48"/><b>then </b><i><font color="Green" title="E44">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="combgras.html#E1:48"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></i>;<br/>
<a NAME="E9:48"/><i><font color="Green" title="E45">A8</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> <span class="p2">(<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>,<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> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 </span>}</span> 
 
<b>by </b><i><a class="txt" href="combgras.html#E7:48"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></i>;<br/>
<a NAME="E10:48"/><i><font color="Green" title="E46">A9</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> <span class="p2">(<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>,<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> <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>}</span> 
 
<b>by </b><i><a class="txt" href="combgras.html#E7:48"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></i>;<br/>
<a NAME="E11:48"/><i><font color="Green" title="E47">A10</font></i>: 
( 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; <font color="Maroon" title="c3">F</font> is <a href="combgras.html#V6" title="COMBGRAS:attr.6">incidence_preserving</a> &amp;  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<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>,<font color="Maroon" title="c2">X</font></span>)</span> &amp;  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<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>,<font color="Maroon" title="c2">X</font></span>)</span> &amp;  <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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<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>,<font color="Maroon" title="c2">X</font></span>)</span> &amp;  <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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<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>,<font color="Maroon" title="c2">X</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="combgras.html#E2:48"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a>, <a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></i>;<br/>
<a NAME="E12:48"/><b>then </b><i><font color="Green" title="E48">A11</font></i>: 
( 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_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E13:48"/><i><font color="Green" title="E49">A12</font></i>: 
the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<a NAME="E1:48_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> holds <br/>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="Olive" title="b1">x</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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <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"> ( <font color="Maroon" title="c6">x</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> implies 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="c6">x</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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<b>assume </b><a NAME="E1:48_1_1"/>
<font color="Maroon" title="c6">x</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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="c6">x</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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">A</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <span class="p2">(<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>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E3:48_1_1"/><i><font color="Green" title="E50">A13</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c8">A1</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="E5:48_1_1"/><i><font color="Green" title="E51">A14</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A1</font>
 <b>by </b><i><a class="txt" href="combgras.html#E8:48"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="combgras.html#E9:48"><i><font color="Green" title="E45">A8</font></i></a>, <a class="txt" href="combgras.html#E3:48_1_1"><i><font color="Green" title="E50">A13</font></i></a></i>;<br/>
<a NAME="E6:48_1_1"/>
<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="c5">f</font></span>)</span> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c8">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c8">A1</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E1:48"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="combgras.html#E7:48"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="combgras.html#D14" target="_self" title="COMBGRAS:def.14">Def14</a></i>;<br/>
<a NAME="E7:48_1_1"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c7">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> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c7">A</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E3:48"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="combgras.html#E4:48"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="combgras.html#E7:48"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="combgras.html#E3:48_1_1"><i><font color="Green" title="E50">A13</font></i></a>, <a class="txt" href="combgras.html#E5:48_1_1"><i><font color="Green" title="E51">A14</font></i></a>, <a class="ref" href="combgras.html#D14" target="_self" title="COMBGRAS:def.14">Def14</a></i>;<br/>
<b>hence </b><a NAME="E8:48_1_1"/>
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="c6">x</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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="combgras.html#E3:48_1_1"><i><font color="Green" title="E50">A13</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="E2:48_1"/>
the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="combgras.html#E11:48"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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="E14:48"/>
the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<a NAME="E1:48_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> holds <br/>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="Olive" title="b1">x</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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <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"> ( <font color="Maroon" title="c6">x</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> implies 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="c6">x</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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<b>assume </b><a NAME="E1:48_2_1"/>
<font color="Maroon" title="c6">x</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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="c6">x</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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">A</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> <span class="p2">(<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>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E3:48_2_1"/><i><font color="Green" title="E50">A15</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font>
 ;<br/>
<a NAME="E4:48_2_1"/>
<font color="Maroon" title="c6">x</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> <span class="p2">(<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>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="combgras.html#E3:48_2_1"><i><font color="Green" title="E50">A15</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">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="E6:48_2_1"/><i><font color="Green" title="E51">A16</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A11</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c8">A11</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="combgras.html#E10:48"><i><font color="Green" title="E46">A9</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">B1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:48_2_1"/><i><font color="Green" title="E52">A17</font></i>: 
( <font color="Maroon" title="c9">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c9">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="combgras.html#E6:48"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="combgras.html#E6:48_2_1"><i><font color="Green" title="E51">A16</font></i></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">C1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:48_2_1"/><i><font color="Green" title="E53">A18</font></i>: 
( <font color="Maroon" title="c10">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">B1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 <b>by </b><i><a class="txt" href="combgras.html#E6:48"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="combgras.html#E8:48_2_1"><i><font color="Green" title="E52">A17</font></i></a>, <a class="ref" href="card_fil.html#T36" title="CARD_FIL:th.36">CARD_FIL:36</a></i>;<br/>
<a NAME="E11:48_2_1"/><i><font color="Green" title="E54">A19</font></i>: 
( <font color="Maroon" title="c9">B1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c10">C1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c6">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c10">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E6:48"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="combgras.html#E6:48_2_1"><i><font color="Green" title="E51">A16</font></i></a>, <a class="txt" href="combgras.html#E8:48_2_1"><i><font color="Green" title="E52">A17</font></i></a>, <a class="txt" href="combgras.html#E10:48_2_1"><i><font color="Green" title="E53">A18</font></i></a>, <a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:48_2_1"/><b>then </b><i><font color="Green" title="E55">A20</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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> &amp; <font color="Maroon" title="c10">C1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">B1</font> &amp; <font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">B1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c10">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">B1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E6:48_2_1"><i><font color="Green" title="E51">A16</font></i></a>, <a class="txt" href="combgras.html#E8:48_2_1"><i><font color="Green" title="E52">A17</font></i></a>, <a class="txt" href="combgras.html#E10:48_2_1"><i><font color="Green" title="E53">A18</font></i></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a>, <a class="ref" href="xboole_1.html#T85" title="XBOOLE_1:th.85">XBOOLE_1:85</a></i>;<br/>
<a NAME="E13:48_2_1"/><b>then </b><i><font color="Green" title="E56">A21</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">B1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">B1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c9">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c9">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">B1</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E6:48_2_1"><i><font color="Green" title="E51">A16</font></i></a>, <a class="txt" href="combgras.html#E8:48_2_1"><i><font color="Green" title="E52">A17</font></i></a>, <a class="txt" href="combgras.html#E10:48_2_1"><i><font color="Green" title="E53">A18</font></i></a>, <a class="txt" href="combgras.html#E11:48_2_1"><i><font color="Green" title="E54">A19</font></i></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E14:48_2_1"/><b>then </b><i><font color="Green" title="E57">A22</font></i>: 
( <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> <span class="p2">(<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>,<font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Maroon" title="c10">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">B1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <span class="p2">(<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>,<font color="Maroon" title="c2">X</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="combgras.html#E9:48"><i><font color="Green" title="E45">A8</font></i></a>, <a class="txt" href="combgras.html#E8:48_2_1"><i><font color="Green" title="E52">A17</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">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> <span class="p2">(<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>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E16:48_2_1"/><i><font color="Green" title="E58">A23</font></i>: 
<font color="Maroon" title="c11">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B1</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c12">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> <span class="p2">(<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>,<font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E18:48_2_1"/><i><font color="Green" title="E59">A24</font></i>: 
<font color="Maroon" title="c12">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">B1</font></span>)</span>
 <b>by </b><i><a class="txt" href="combgras.html#E14:48_2_1"><i><font color="Green" title="E57">A22</font></i></a></i>;<br/>
<a NAME="E19:48_2_1"/>
<font color="Maroon" title="c9">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">B1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E20:48_2_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="c6">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">B1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c9">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T78" title="CARD_1:th.78">CARD_1:78</a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E21:48_2_1"/><b>then </b><i><font color="Green" title="E60">A25</font></i>: 
<font color="Maroon" title="c11">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">b2</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E12:48_2_1"><i><font color="Green" title="E55">A20</font></i></a>, <a class="txt" href="combgras.html#E16:48_2_1"><i><font color="Green" title="E58">A23</font></i></a>, <a class="txt" href="combgras.html#E18:48_2_1"><i><font color="Green" title="E59">A24</font></i></a>, <a class="ref" href="xboole_1.html#T11" title="XBOOLE_1:th.11">XBOOLE_1:11</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E22:48_2_1"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">b2</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> 2 &amp; <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 <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="c11">b1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">b2</font></span>)</span> &amp; <font color="Maroon" title="c12">b2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c11">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c12">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">A</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E6:48"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="combgras.html#E7:48"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="combgras.html#E3:48_2_1"><i><font color="Green" title="E50">A15</font></i></a>, <a class="txt" href="combgras.html#E6:48_2_1"><i><font color="Green" title="E51">A16</font></i></a>, <a class="txt" href="combgras.html#E8:48_2_1"><i><font color="Green" title="E52">A17</font></i></a>, <a class="txt" href="combgras.html#E12:48_2_1"><i><font color="Green" title="E55">A20</font></i></a>, <a class="txt" href="combgras.html#E13:48_2_1"><i><font color="Green" title="E56">A21</font></i></a>, <a class="txt" href="combgras.html#E16:48_2_1"><i><font color="Green" title="E58">A23</font></i></a>, <a class="txt" href="combgras.html#E18:48_2_1"><i><font color="Green" title="E59">A24</font></i></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a></i>;<br/>
<a NAME="E23:48_2_1"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">b2</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> 2 &amp; <font color="Maroon" title="c11">b1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">b2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c11">b1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">b2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b1</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="c7">A</font> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c12">b2</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="c7">A</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E11:48"><i><font color="Green" title="E47">A10</font></i></a>, <a class="txt" href="combgras.html#E8:48_2_1"><i><font color="Green" title="E52">A17</font></i></a>, <a class="txt" href="combgras.html#E11:48_2_1"><i><font color="Green" title="E54">A19</font></i></a>, <a class="txt" href="combgras.html#E12:48_2_1"><i><font color="Green" title="E55">A20</font></i></a>, <a class="txt" href="combgras.html#E16:48_2_1"><i><font color="Green" title="E58">A23</font></i></a>, <a class="txt" href="combgras.html#E18:48_2_1"><i><font color="Green" title="E59">A24</font></i></a>, <a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">Def8</a>, <a class="ref" href="finset_1.html#T14" title="FINSET_1:th.14">FINSET_1:14</a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E24:48_2_1"/><b>then </b><i><font color="Green" title="E61">A26</font></i>: 
( <font color="Maroon" title="c11">b1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c11">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c12">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">b2</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">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> <span class="p2">(<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>,<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="c12">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> <span class="p2">(<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>,<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="c7">A</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> <span class="p2">(<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>,<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">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c12">b2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c7">A</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E7:48"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="combgras.html#E6:48_2_1"><i><font color="Green" title="E51">A16</font></i></a>, <a class="txt" href="combgras.html#E11:48_2_1"><i><font color="Green" title="E54">A19</font></i></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a>, <a class="ref" href="relat_1.html#T153" title="RELAT_1:th.153">RELAT_1:153</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">B11</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="E26:48_2_1"/><i><font color="Green" title="E62">A27</font></i>: 
( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B11</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c13">B11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="combgras.html#E9:48"><i><font color="Green" title="E45">A8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">B12</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="E28:48_2_1"/><i><font color="Green" title="E63">A28</font></i>: 
( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c12">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B12</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c14">B12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="combgras.html#E9:48"><i><font color="Green" title="E45">A8</font></i></a>, <a class="txt" href="combgras.html#E24:48_2_1"><i><font color="Green" title="E61">A26</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">L1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E30:48_2_1"/><i><font color="Green" title="E64">A29</font></i>: 
( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">L1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c15">L1</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="combgras.html#E10:48"><i><font color="Green" title="E46">A9</font></i></a>, <a class="txt" href="combgras.html#E24:48_2_1"><i><font color="Green" title="E61">A26</font></i></a></i>;<br/>
<a NAME="E31:48_2_1"/><i><font color="Green" title="E65">A30</font></i>: 
( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c12">b2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c12">b2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c7">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="txt" href="combgras.html#E6:48"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="combgras.html#E11:48"><i><font color="Green" title="E47">A10</font></i></a>, <a class="txt" href="combgras.html#E12:48"><i><font color="Green" title="E48">A11</font></i></a>, <a class="txt" href="combgras.html#E21:48_2_1"><i><font color="Green" title="E60">A25</font></i></a>, <a class="txt" href="combgras.html#E24:48_2_1"><i><font color="Green" title="E61">A26</font></i></a>, <a class="txt" href="combgras.html#E30:48_2_1"><i><font color="Green" title="E64">A29</font></i></a>, <a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E32:48_2_1"/><b>then </b>
( <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 <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="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c12">b2</font></span>)</span></span>)</span> &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="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c12">b2</font></span>)</span></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> 2 )
 
<b>by </b><i><a class="txt" href="combgras.html#E26:48_2_1"><i><font color="Green" title="E62">A27</font></i></a>, <a class="txt" href="combgras.html#E28:48_2_1"><i><font color="Green" title="E63">A28</font></i></a>, <a class="txt" href="combgras.html#E30:48_2_1"><i><font color="Green" title="E64">A29</font></i></a>, <a class="ref" href="combgras.html#T1" target="_self" title="COMBGRAS:th.1">Th1</a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E33:48_2_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="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c12">b2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c12">b2</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="txt" href="combgras.html#E31:48_2_1"><i><font color="Green" title="E65">A30</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E34:48_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c12">b2</font></span>)</span> <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="c7">A</font> &amp; <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c12">b2</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> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c12">b2</font> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b1</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> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b1</font> &amp; <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c11">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c11">b1</font> &amp; <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c12">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c12">b2</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E7:48"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="combgras.html#E13:48"><i><font color="Green" title="E49">A12</font></i></a>, <a class="txt" href="combgras.html#E3:48_2_1"><i><font color="Green" title="E50">A15</font></i></a>, <a class="txt" href="combgras.html#E30:48_2_1"><i><font color="Green" title="E64">A29</font></i></a>, <a class="txt" href="combgras.html#E31:48_2_1"><i><font color="Green" title="E65">A30</font></i></a>, <a class="ref" href="combgras.html#D14" target="_self" title="COMBGRAS:def.14">Def14</a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a></i>;<br/>
<b>hence </b><a NAME="E35:48_2_1"/>
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="c6">x</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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="combgras.html#E3:48_2_1"><i><font color="Green" title="E50">A15</font></i></a>, <a class="txt" href="combgras.html#E24:48_2_1"><i><font color="Green" title="E61">A26</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="E2:48_2"/>
the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <span class="p2">(<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>,<font color="Maroon" title="c5">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="combgras.html#E11:48"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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="E15:48"/>
 <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> <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>,<font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="combgras.html#E13:48"><i><font color="Green" title="E49">A12</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>
