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

<b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<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> ;<br/>



<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E37">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="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">X</font> )
 ;<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>;<br/>

<b>assume </b><a NAME="E2:47"/><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c3">F</font> is <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a>
 ;<br/>

<a NAME="E3:47"/><i><font color="Green" title="E39">A3</font></i>: 
( <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> 2 <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="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> 2 &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &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; <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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="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> 2 )
 
<b>by </b><i><a class="txt" href="combgras.html#E1:47"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E4:47"/><b>then </b>
( <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="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; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E5:47"/><b>then </b><i><font color="Green" title="E40">A4</font></i>: 
( <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">X</font> &amp; 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <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> 2 <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</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="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</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#E1:47"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:47"/><b>then </b><i><font color="Green" title="E41">A5</font></i>: 
the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="combgras.html#E1:47"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></i>;<br/>
<a NAME="E7:47"/><i><font color="Green" title="E42">A6</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#E1:47"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></i>;<br/>
<a NAME="E8:47"/><i><font color="Green" title="E43">A7</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#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $2 <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="b1">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> );<br/>
<a NAME="E9:47"/><i><font color="Green" title="E44">A8</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E3:47_1"/><i><font color="Green" title="E45">A9</font></i>: 
( <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 <b>by </b><i><a class="txt" href="combgras.html#E6:47"><i><font color="Green" title="E41">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c4">e</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="combgras.html#E3:47"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="combgras.html#E3:47_1"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a></i>;<br/>
<b>take </b><font color="Maroon" title="c7">u</font> =  <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="Maroon" title="c6">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>;<br/>

<b>thus </b><a NAME="E5:47_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c7">u</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">Hp</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E11:47"/><i><font color="Green" title="E45">A10</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">Hp</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> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 <b>and </b><br/><a NAME="E12:47"/><i><font color="Green" title="E46">A11</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c4">Hp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>]
 <b>from </b><i><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="combgras.html#E9:47"><i><font color="Green" title="E44">A8</font></i></a>);<br/></i>
<a NAME="E13:47"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">Hp</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:47_2"/><i><font color="Green" title="E47">A12</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">Hp</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:47_2"/><i><font color="Green" title="E48">A13</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">Hp</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Hp</font> <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#E1:47_2"><i><font color="Green" title="E47">A12</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">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> <b> such that </b><br/><a NAME="E5:47_2"/><i><font color="Green" title="E49">A14</font></i>: 
( <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c5">y</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="Maroon" title="c7">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> )
 <b>by </b><i><a class="txt" href="combgras.html#E11:47"><i><font color="Green" title="E45">A10</font></i></a>, <a class="txt" href="combgras.html#E12:47"><i><font color="Green" title="E46">A11</font></i></a>, <a class="txt" href="combgras.html#E3:47_2"><i><font color="Green" title="E48">A13</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">x1</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="E7:47_2"/><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="c8">x1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c8">x1</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#E6:47"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="combgras.html#E11:47"><i><font color="Green" title="E45">A10</font></i></a>, <a class="txt" href="combgras.html#E3:47_2"><i><font color="Green" title="E48">A13</font></i></a></i>;<br/>
<a NAME="E8:47_2"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E5:47_2"><i><font color="Green" title="E49">A14</font></i></a>, <a class="txt" href="combgras.html#E7:47_2"><i><font color="Green" title="E50">A15</font></i></a></i>;<br/>
<a NAME="E9:47_2"/><b>then </b>
 <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c2">X</font>,<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> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>
 
<b>by </b><i><a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="combgras.html#T29" target="_self" title="COMBGRAS:th.29">Th29</a></i>;<br/>
<a NAME="E10:47_2"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p1">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c2">X</font>,<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></span>)</span> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>
 
<b>by </b><i><a class="txt" href="combgras.html#E2:47"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="combgras.html#T23" target="_self" title="COMBGRAS:th.23">Th23</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">S</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E12:47_2"/><i><font color="Green" title="E51">A16</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c9">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p1">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c2">X</font>,<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</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">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">C</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="combgras.html#D4" target="_self" title="COMBGRAS:def.4">Def4</a></i>;<br/>
<a NAME="E13:47_2"/>
( <font color="Maroon" title="c9">S</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="Maroon" title="c7">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> &amp; <font color="Maroon" title="c9">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="combgras.html#E6:47"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="combgras.html#E12:47_2"><i><font color="Green" title="E51">A16</font></i></a>, <a class="ref" href="combgras.html#T26" target="_self" title="COMBGRAS:th.26">Th26</a></i>;<br/>
<b>hence </b><a NAME="E14:47_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="combgras.html#E5:47_2"><i><font color="Green" title="E49">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">Hp</font> = <font color="Maroon" title="c4">Hp</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>,the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="combgras.html#E11:47"><i><font color="Green" title="E45">A10</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E15:47"/>
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>
 
<b>by </b><i><a class="txt" href="combgras.html#E2:47"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a></i>;<br/>
<a NAME="E16:47"/><b>then </b><i><font color="Green" title="E47">A17</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#V2" title="FUNCT_2:attr.2">onto</a> &amp; the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E17:47"/><b>then </b><i><font color="Green" title="E48">A18</font></i>: 
(  <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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> <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> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">Hl</font> = the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 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>,the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="combgras.html#E7:47"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="combgras.html#E8:47"><i><font color="Green" title="E43">A7</font></i></a></i>;<br/>
<b>take </b><font color="Maroon" title="c7">H</font> =  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(# <font color="Maroon" title="c5">Hp</font>,<font color="Maroon" title="c6">Hl</font> #);<br/>

<a NAME="E19:47"/><i><font color="Green" title="E49">A19</font></i>: 
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="c5">Hp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">A</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c9">B</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:47_3"/><i><font color="Green" title="E50">A20</font></i>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B</font>
 ;<br/>

<a NAME="E2:47_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c5">Hp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">A</font>]
 
<b>by </b><i><a class="txt" href="combgras.html#E12:47"><i><font color="Green" title="E46">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:47_3"/>
<font color="Maroon" title="c5">Hp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">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="Maroon" title="c9">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>
 <b>by </b><i><a class="txt" href="combgras.html#E1:47_3"><i><font color="Green" title="E50">A20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:47"/>
<font color="Maroon" title="c7">H</font> is <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4"><b>proof </b></a><div class="add">

<a NAME="E1:47_4"/><i><font color="Green" title="E50">A21</font></i>: 
the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c7">H</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 
<b>by </b><i><a class="txt" href="combgras.html#E2:47"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="combgras.html#E7:47"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="combgras.html#E8:47"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a></i>;<br/>
<a NAME="E2:47_4"/><i><font color="Green" title="E51">A22</font></i>: 
the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c7">H</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:47_4_1"/><i><font color="Green" title="E52">A23</font></i>: 
<font color="Maroon" title="c5">Hp</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:47_4_1_1"/><i><font color="Green" title="E53">A24</font></i>: 
( <font color="Maroon" title="c8">x1</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> <font color="Maroon" title="c5">Hp</font> &amp; <font color="Maroon" title="c9">x2</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> <font color="Maroon" title="c5">Hp</font> &amp; <font color="Maroon" title="c5">Hp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Hp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x2</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c10">X1</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> <b> such that </b><br/><a NAME="E3:47_4_1_1"/><i><font color="Green" title="E54">A25</font></i>: 
( <font color="Maroon" title="c10">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x1</font> &amp; <font color="Maroon" title="c5">Hp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</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="Maroon" title="c10">X1</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> )
 <b>by </b><i><a class="txt" href="combgras.html#E12:47"><i><font color="Green" title="E46">A11</font></i></a>, <a class="txt" href="combgras.html#E1:47_4_1_1"><i><font color="Green" title="E53">A24</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">X2</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> <b> such that </b><br/><a NAME="E5:47_4_1_1"/><i><font color="Green" title="E55">A26</font></i>: 
( <font color="Maroon" title="c11">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font> &amp; <font color="Maroon" title="c5">Hp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x2</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="Maroon" title="c11">X2</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> )
 <b>by </b><i><a class="txt" href="combgras.html#E12:47"><i><font color="Green" title="E46">A11</font></i></a>, <a class="txt" href="combgras.html#E1:47_4_1_1"><i><font color="Green" title="E53">A24</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">x11</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="E7:47_4_1_1"/><i><font color="Green" title="E56">A27</font></i>: 
( <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x11</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c12">x11</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#E6:47"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="combgras.html#E11:47"><i><font color="Green" title="E45">A10</font></i></a>, <a class="txt" href="combgras.html#E1:47_4_1_1"><i><font color="Green" title="E53">A24</font></i></a></i>;<br/>
<a NAME="E8:47_4_1_1"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c10">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; <font color="Maroon" title="c10">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:47_4_1_1"><i><font color="Green" title="E54">A25</font></i></a>, <a class="txt" href="combgras.html#E7:47_4_1_1"><i><font color="Green" title="E56">A27</font></i></a></i>;<br/>
<a NAME="E9:47_4_1_1"/><b>then </b>
 <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c10">X1</font>,<font color="Maroon" title="c2">X</font>,<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> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>
 
<b>by </b><i><a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="combgras.html#T29" target="_self" title="COMBGRAS:th.29">Th29</a></i>;<br/>
<a NAME="E10:47_4_1_1"/><b>then </b><i><font color="Green" title="E57">A28</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p1">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c10">X1</font>,<font color="Maroon" title="c2">X</font>,<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></span>)</span> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>
 
<b>by </b><i><a class="txt" href="combgras.html#E2:47"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="combgras.html#T23" target="_self" title="COMBGRAS:th.23">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">x12</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="E12:47_4_1_1"/><i><font color="Green" title="E58">A29</font></i>: 
( <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x12</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c13">x12</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#E6:47"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="combgras.html#E11:47"><i><font color="Green" title="E45">A10</font></i></a>, <a class="txt" href="combgras.html#E1:47_4_1_1"><i><font color="Green" title="E53">A24</font></i></a></i>;<br/>
<a NAME="E13:47_4_1_1"/><i><font color="Green" title="E59">A30</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c11">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; <font color="Maroon" title="c11">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E5:47_4_1_1"><i><font color="Green" title="E55">A26</font></i></a>, <a class="txt" href="combgras.html#E12:47_4_1_1"><i><font color="Green" title="E58">A29</font></i></a></i>;<br/>
<a NAME="E14:47_4_1_1"/><b>then </b>
 <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c11">X2</font>,<font color="Maroon" title="c2">X</font>,<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> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>
 
<b>by </b><i><a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="combgras.html#T29" target="_self" title="COMBGRAS:th.29">Th29</a></i>;<br/>
<a NAME="E15:47_4_1_1"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p1">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c11">X2</font>,<font color="Maroon" title="c2">X</font>,<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></span>)</span> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>
 
<b>by </b><i><a class="txt" href="combgras.html#E2:47"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="combgras.html#T23" target="_self" title="COMBGRAS:th.23">Th23</a></i>;<br/>
<a NAME="E16:47_4_1_1"/><b>then </b><i><font color="Green" title="E60">A31</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p1">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c10">X1</font>,<font color="Maroon" title="c2">X</font>,<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p1">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c11">X2</font>,<font color="Maroon" title="c2">X</font>,<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></span>)</span>
 
<b>by </b><i><a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="combgras.html#E1:47_4_1_1"><i><font color="Green" title="E53">A24</font></i></a>, <a class="txt" href="combgras.html#E3:47_4_1_1"><i><font color="Green" title="E54">A25</font></i></a>, <a class="txt" href="combgras.html#E5:47_4_1_1"><i><font color="Green" title="E55">A26</font></i></a>, <a class="txt" href="combgras.html#E10:47_4_1_1"><i><font color="Green" title="E57">A28</font></i></a>, <a class="ref" href="combgras.html#T28" target="_self" title="COMBGRAS:th.28">Th28</a></i>;<br/>
<a NAME="E17:47_4_1_1"/>
(  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c10">X1</font>,<font color="Maroon" title="c2">X</font>,<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">X1</font> &amp;  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c11">X2</font>,<font color="Maroon" title="c2">X</font>,<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">X2</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="combgras.html#E3:47_4_1_1"><i><font color="Green" title="E54">A25</font></i></a>, <a class="txt" href="combgras.html#E7:47_4_1_1"><i><font color="Green" title="E56">A27</font></i></a>, <a class="txt" href="combgras.html#E13:47_4_1_1"><i><font color="Green" title="E59">A30</font></i></a>, <a class="ref" href="combgras.html#T30" target="_self" title="COMBGRAS:th.30">Th30</a></i>;<br/>
<b>hence </b><a NAME="E18:47_4_1_1"/>
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font>
 <b>by </b><i><a class="txt" href="combgras.html#E16:47"><i><font color="Green" title="E47">A17</font></i></a>, <a class="txt" href="combgras.html#E3:47_4_1_1"><i><font color="Green" title="E54">A25</font></i></a>, <a class="txt" href="combgras.html#E5:47_4_1_1"><i><font color="Green" title="E55">A26</font></i></a>, <a class="txt" href="combgras.html#E16:47_4_1_1"><i><font color="Green" title="E60">A31</font></i></a>, <a class="ref" href="combgras.html#T6" target="_self" title="COMBGRAS:th.6">Th6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:47_4_1"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Hp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_4_1_2"/><i><font color="Green" title="E53">A32</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">Y1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E3:47_4_1_2"/><i><font color="Green" title="E54">A33</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Y1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c9">Y1</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#E6:47"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="combgras.html#E1:47_4_1_2"><i><font color="Green" title="E53">A32</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c8">y</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="combgras.html#E3:47"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="combgras.html#E3:47_4_1_2"><i><font color="Green" title="E54">A33</font></i></a>, <a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a></i>;<br/>
<a NAME="E5:47_4_1_2"/><i><font color="Green" title="E55">A34</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<b>by </b><i><a class="txt" href="combgras.html#E3:47_4_1_2"><i><font color="Green" title="E54">A33</font></i></a></i>;<br/>
<a NAME="E6:47_4_1_2"/><b>then </b>
 <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c2">X</font>,<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> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>
 
<b>by </b><i><a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="combgras.html#T29" target="_self" title="COMBGRAS:th.29">Th29</a></i>;<br/>
<a NAME="E7:47_4_1_2"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <span class="p1">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c2">X</font>,<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></span>)</span> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>
 
<b>by </b><i><a class="txt" href="combgras.html#E2:47"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="combgras.html#T23" target="_self" title="COMBGRAS:th.23">Th23</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">S</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E9:47_4_1_2"/><i><font color="Green" title="E56">A35</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c11">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <span class="p1">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c2">X</font>,<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></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 &amp; <font color="Maroon" title="c11">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="combgras.html#D4" target="_self" title="COMBGRAS:def.4">Def4</a></i>;<br/>
<a NAME="E10:47_4_1_2"/><i><font color="Green" title="E57">A36</font></i>: 
( <font color="Maroon" title="c11">S</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#K5" title="COMBGRAS:func.5">"</a> <span class="p2">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c10">y</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> &amp; <font color="Maroon" title="c11">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="combgras.html#E6:47"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="combgras.html#E9:47_4_1_2"><i><font color="Green" title="E56">A35</font></i></a>, <a class="ref" href="combgras.html#T26" target="_self" title="COMBGRAS:th.26">Th26</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">S</font> = <font color="Maroon" title="c11">S</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="combgras.html#E3:47"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="combgras.html#E9:47_4_1_2"><i><font color="Green" title="E56">A35</font></i></a>, <a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a></i>;<br/>
<a NAME="E12:47_4_1_2"/>
 <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c12">S</font>,<font color="Maroon" title="c2">X</font>,<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>  <a href="combgras.html#K6" title="COMBGRAS:func.6">^^</a> <font color="Maroon" title="c12">S</font>,<font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="combgras.html#E9:47_4_1_2"><i><font color="Green" title="E56">A35</font></i></a>, <a class="ref" href="combgras.html#D13" target="_self" title="COMBGRAS:def.13">Def13</a></i>;<br/>
<a NAME="E13:47_4_1_2"/><b>then </b>
( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <span class="p1">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c2">X</font>,<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c12">S</font>,<font color="Maroon" title="c2">X</font>,<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">S<sub>1</sub></font>[<font color="Maroon" title="c12">S</font>,<font color="Maroon" title="c5">Hp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">S</font>] )
 
<b>by </b><i><a class="txt" href="combgras.html#E12:47"><i><font color="Green" title="E46">A11</font></i></a>, <a class="txt" href="combgras.html#E9:47_4_1_2"><i><font color="Green" title="E56">A35</font></i></a>, <a class="txt" href="combgras.html#E10:47_4_1_2"><i><font color="Green" title="E57">A36</font></i></a></i>;<br/>
<a NAME="E14:47_4_1_2"/><b>then </b>
<font color="Maroon" title="c5">Hp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">S</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"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c2">X</font>,<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></span>)</span>
 
<b>by </b><i><a class="txt" href="combgras.html#E17:47"><i><font color="Green" title="E48">A18</font></i></a>, <a class="ref" href="funct_1.html#T147" title="FUNCT_1:th.147">FUNCT_1:147</a></i>;<br/>
<a NAME="E15:47_4_1_2"/><b>then </b>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Hp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">S</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="combgras.html#E5:47_4_1_2"><i><font color="Green" title="E55">A34</font></i></a>, <a class="ref" href="combgras.html#T30" target="_self" title="COMBGRAS:th.30">Th30</a></i>;<br/>
<b>hence </b><a NAME="E16:47_4_1_2"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Hp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 <b>by </b><i><a class="txt" href="combgras.html#E10:47_4_1_2"><i><font color="Green" title="E57">A36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:47_4_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">Hp</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> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T16" title="FUNCT_2:th.16">FUNCT_2:16</a></i>;<br/>
<a NAME="E4:47_4_1"/><b>then </b>
<font color="Maroon" title="c5">Hp</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#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:47_4_1"/>
the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c7">H</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 <b>by </b><i><a class="txt" href="combgras.html#E1:47_4_1"><i><font color="Green" title="E52">A23</font></i></a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:47_4"/>
<font color="Maroon" title="c7">H</font> is <a href="combgras.html#V6" title="COMBGRAS:attr.6">incidence_preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">A1</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">COMBGRAS:def 8</a><br/><b>let </b><font color="Maroon" title="c9">L1</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>;<br/>



<a NAME="E1:47_4_2"/>
<font color="Maroon" title="c8">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c10">a1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E3:47_4_2"/><i><font color="Green" title="E52">A37</font></i>: 
( <font color="Maroon" title="c10">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c10">a1</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#E6:47"><i><font color="Green" title="E41">A5</font></i></a></i>;<br/>
<a NAME="E4:47_4_2"/>
<font color="Maroon" title="c9">L1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c11">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="E6:47_4_2"/><i><font color="Green" title="E53">A38</font></i>: 
( <font color="Maroon" title="c11">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">L1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c11">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="combgras.html#E7:47"><i><font color="Green" title="E42">A6</font></i></a></i>;<br/>
<a NAME="E7:47_4_2"/>
<font color="Maroon" title="c9">L1</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>
 
<b>by </b><i><a class="txt" href="combgras.html#E8:47"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="combgras.html#E6:47_4_2"><i><font color="Green" title="E53">A38</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">L11</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="E9:47_4_2"/><i><font color="Green" title="E54">A39</font></i>: 
<font color="Maroon" title="c12">L11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">L1</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">a1</font> = <font color="Maroon" title="c10">a1</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="combgras.html#E3:47"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="combgras.html#E3:47_4_2"><i><font color="Green" title="E52">A37</font></i></a>, <a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a></i>;<br/>
<a NAME="E11:47_4_2"/><i><font color="Green" title="E55">A40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">A1</font>,<font color="Maroon" title="c5">Hp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">A1</font>]
 
<b>by </b><i><a class="txt" href="combgras.html#E12:47"><i><font color="Green" title="E46">A11</font></i></a></i>;<br/>
<a NAME="E12:47_4_2"/><i><font color="Green" title="E56">A41</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c13">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<b>by </b><i><a class="txt" href="combgras.html#E3:47_4_2"><i><font color="Green" title="E52">A37</font></i></a></i>;<br/>
<a NAME="E13:47_4_2"/><b>then </b><i><font color="Green" title="E57">A42</font></i>: 
 <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c2">X</font>,<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>  <a href="combgras.html#K6" title="COMBGRAS:func.6">^^</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="combgras.html#D13" target="_self" title="COMBGRAS:def.13">Def13</a></i>;<br/>
<a NAME="E14:47_4_2"/><i><font color="Green" title="E58">A43</font></i>: 
( <font color="Maroon" title="c8">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L1</font> implies <font color="Maroon" title="c7">H</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c8">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">H</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c9">L1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_4_2_1"/>
<font color="Maroon" title="c8">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L1</font>
 ;<br/>

<a NAME="E2:47_4_2_1"/><b>then </b>
<font color="Maroon" title="c8">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">L1</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E1:47"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></i>;<br/>
<a NAME="E3:47_4_2_1"/><b>then </b>
<font color="Maroon" title="c9">L1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c2">X</font>,<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>
 
<b>by </b><i><a class="txt" href="combgras.html#E3:47_4_2"><i><font color="Green" title="E52">A37</font></i></a>, <a class="txt" href="combgras.html#E6:47_4_2"><i><font color="Green" title="E53">A38</font></i></a>, <a class="txt" href="combgras.html#E13:47_4_2"><i><font color="Green" title="E57">A42</font></i></a></i>;<br/>
<a NAME="E4:47_4_2_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="c12">L11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p1">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c2">X</font>,<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></span>)</span>
 
<b>by </b><i><a class="txt" href="combgras.html#E17:47"><i><font color="Green" title="E48">A18</font></i></a>, <a class="txt" href="combgras.html#E9:47_4_2"><i><font color="Green" title="E54">A39</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E5:47_4_2_1"/><b>then </b>
(  <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="Maroon" title="c13">a1</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 href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c12">L11</font> &amp; <font color="Maroon" title="c7">H</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c9">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c12">L11</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E9:47_4_2"><i><font color="Green" title="E54">A39</font></i></a>, <a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></i>;<br/>
<b>hence </b><a NAME="E6:47_4_2_1"/>
<font color="Maroon" title="c7">H</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c8">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">H</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c9">L1</font>
 <b>by </b><i><a class="txt" href="combgras.html#E1:47"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="combgras.html#E3:47_4_2"><i><font color="Green" title="E52">A37</font></i></a>, <a class="txt" href="combgras.html#E11:47_4_2"><i><font color="Green" title="E55">A40</font></i></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:47_4_2"/>
( <font color="Maroon" title="c7">H</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c8">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">H</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c9">L1</font> implies <font color="Maroon" title="c8">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_4_2_2"/>
<font color="Maroon" title="c7">H</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c8">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">H</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c9">L1</font>
 ;<br/>

<a NAME="E2:47_4_2_2"/><b>then </b><i><font color="Green" title="E59">A44</font></i>: 
<font color="Maroon" title="c7">H</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c8">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">H</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c9">L1</font>
 
<b>by </b><i><a class="txt" href="combgras.html#E1:47"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></i>;<br/>
<a NAME="E3:47_4_2_2"/>
<font color="Maroon" title="c7">H</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c8">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c14">ha1</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="E5:47_4_2_2"/><i><font color="Green" title="E60">A45</font></i>: 
( <font color="Maroon" title="c14">ha1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">H</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c8">A1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c14">ha1</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#E6:47"><i><font color="Green" title="E41">A5</font></i></a></i>;<br/>
<a NAME="E6:47_4_2_2"/>
<font color="Maroon" title="c7">H</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c9">L1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c15">hl1</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="E8:47_4_2_2"/><i><font color="Green" title="E61">A46</font></i>: 
( <font color="Maroon" title="c15">hl1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">H</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c9">L1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c15">hl1</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#E7:47"><i><font color="Green" title="E42">A6</font></i></a></i>;<br/>
<a NAME="E9:47_4_2_2"/>
 <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c2">X</font>,<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> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>
 
<b>by </b><i><a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="combgras.html#E12:47_4_2"><i><font color="Green" title="E56">A41</font></i></a>, <a class="ref" href="combgras.html#T29" target="_self" title="COMBGRAS:th.29">Th29</a></i>;<br/>
<a NAME="E10:47_4_2_2"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p1">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c2">X</font>,<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></span>)</span> is <a href="combgras.html#V3" title="COMBGRAS:attr.3">STAR</a>
 
<b>by </b><i><a class="txt" href="combgras.html#E2:47"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="combgras.html#T23" target="_self" title="COMBGRAS:th.23">Th23</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">S</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E12:47_4_2_2"/><i><font color="Green" title="E62">A47</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c16">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p1">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c2">X</font>,<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></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 &amp; <font color="Maroon" title="c16">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="combgras.html#D4" target="_self" title="COMBGRAS:def.4">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">ha1</font> = <font color="Maroon" title="c14">ha1</font>, <font color="Maroon" title="c18">S</font> = <font color="Maroon" title="c16">S</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="combgras.html#E3:47"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="combgras.html#E5:47_4_2_2"><i><font color="Green" title="E60">A45</font></i></a>, <a class="txt" href="combgras.html#E12:47_4_2_2"><i><font color="Green" title="E62">A47</font></i></a>, <a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a></i>;<br/>
<a NAME="E14:47_4_2_2"/><i><font color="Green" title="E63">A48</font></i>: 
(  <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c17">ha1</font>,<font color="Maroon" title="c2">X</font>,<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>  <a href="combgras.html#K6" title="COMBGRAS:func.6">^^</a> <font color="Maroon" title="c17">ha1</font>,<font color="Maroon" title="c2">X</font> &amp;  <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c18">S</font>,<font color="Maroon" title="c2">X</font>,<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>  <a href="combgras.html#K6" title="COMBGRAS:func.6">^^</a> <font color="Maroon" title="c18">S</font>,<font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="combgras.html#E5:47_4_2_2"><i><font color="Green" title="E60">A45</font></i></a>, <a class="txt" href="combgras.html#E12:47_4_2_2"><i><font color="Green" title="E62">A47</font></i></a>, <a class="ref" href="combgras.html#D13" target="_self" title="COMBGRAS:def.13">Def13</a></i>;<br/>
<a NAME="E15:47_4_2_2"/><b>then </b>
( <font color="Maroon" title="c3">F</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c12">L11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c17">ha1</font>,<font color="Maroon" title="c2">X</font>,<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="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p1">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c2">X</font>,<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c18">S</font>,<font color="Maroon" title="c2">X</font>,<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> )
 
<b>by </b><i><a class="txt" href="combgras.html#E9:47_4_2"><i><font color="Green" title="E54">A39</font></i></a>, <a class="txt" href="combgras.html#E2:47_4_2_2"><i><font color="Green" title="E59">A44</font></i></a>, <a class="txt" href="combgras.html#E5:47_4_2_2"><i><font color="Green" title="E60">A45</font></i></a>, <a class="txt" href="combgras.html#E8:47_4_2_2"><i><font color="Green" title="E61">A46</font></i></a>, <a class="txt" href="combgras.html#E12:47_4_2_2"><i><font color="Green" title="E62">A47</font></i></a></i>;<br/>
<a NAME="E16:47_4_2_2"/><b>then </b>
( <font color="Maroon" title="c9">L1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</a> <span class="p1">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c17">ha1</font>,<font color="Maroon" title="c2">X</font>,<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></span>)</span> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</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="Maroon" title="c13">a1</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 href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c2">X</font>,<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;  <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c2">X</font>,<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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</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="Maroon" title="c13">a1</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> &amp; <font color="Maroon" title="c18">S</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="Maroon" title="c13">a1</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> )
 
<b>by </b><i><a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="combgras.html#E16:47"><i><font color="Green" title="E47">A17</font></i></a>, <a class="txt" href="combgras.html#E17:47"><i><font color="Green" title="E48">A18</font></i></a>, <a class="txt" href="combgras.html#E9:47_4_2"><i><font color="Green" title="E54">A39</font></i></a>, <a class="txt" href="combgras.html#E12:47_4_2_2"><i><font color="Green" title="E62">A47</font></i></a>, <a class="ref" href="combgras.html#T30" target="_self" title="COMBGRAS:th.30">Th30</a>, <a class="ref" href="funct_1.html#T146" title="FUNCT_1:th.146">FUNCT_1:146</a>, <a class="ref" href="funct_1.html#T152" title="FUNCT_1:th.152">FUNCT_1:152</a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E17:47_4_2_2"/><b>then </b>
(  <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#K5" title="COMBGRAS:func.5">"</a> <span class="p2">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c17">ha1</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 href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">L1</font> &amp; <font color="Maroon" title="c3">F</font> <a href="combgras.html#K5" title="COMBGRAS:func.5">"</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="Maroon" title="c13">a1</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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c2">X</font>,<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;  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c2">X</font>,<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a1</font> &amp;  <a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c17">ha1</font>,<font color="Maroon" title="c2">X</font>,<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="c3">F</font> <a href="combgras.html#K4" title="COMBGRAS:func.4">.:</a> <span class="p1">(<span class="default"><a href="combgras.html#K7" title="COMBGRAS:func.7">^^</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c2">X</font>,<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></span>)</span> )
 
<b>by </b><i><a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="combgras.html#E3:47_4_2"><i><font color="Green" title="E52">A37</font></i></a>, <a class="txt" href="combgras.html#E11:47_4_2"><i><font color="Green" title="E55">A40</font></i></a>, <a class="txt" href="combgras.html#E5:47_4_2_2"><i><font color="Green" title="E60">A45</font></i></a>, <a class="txt" href="combgras.html#E12:47_4_2_2"><i><font color="Green" title="E62">A47</font></i></a>, <a class="txt" href="combgras.html#E14:47_4_2_2"><i><font color="Green" title="E63">A48</font></i></a>, <a class="ref" href="combgras.html#T30" target="_self" title="COMBGRAS:th.30">Th30</a>, <a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E18:47_4_2_2"/>
<font color="Maroon" title="c8">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L1</font>
 <b>by </b><i><a class="txt" href="combgras.html#E1:47"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="combgras.html#E5:47"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="combgras.html#E3:47_4_2"><i><font color="Green" title="E52">A37</font></i></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:47_4_2"/>
( <font color="Maroon" title="c8">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L1</font> iff <font color="Maroon" title="c7">H</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c8">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">H</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c9">L1</font> )
 <b>by </b><i><a class="txt" href="combgras.html#E14:47_4_2"><i><font color="Green" title="E58">A43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:47_4"/>
<font color="Maroon" title="c7">H</font> is <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a>
 <b>by </b><i><a class="txt" href="combgras.html#E1:47_4"><i><font color="Green" title="E50">A21</font></i></a>, <a class="txt" href="combgras.html#E2:47_4"><i><font color="Green" title="E51">A22</font></i></a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E21:47"/>
( <font color="Maroon" title="c7">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="c7">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="c7">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> ) )
 <b>by </b><i><a class="txt" href="combgras.html#E19:47"><i><font color="Green" title="E49">A19</font></i></a></i>;<br/>


</div>
