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

<span class="kw">let </span><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">X</font> holds <br/>for <font color="Olive" title="b2">s</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Olive" title="b1">X</font> holds   <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Olive" title="b2">s</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font> implies for <font color="Olive" title="b1">s</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font> holds   <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">s</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:50"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font> holds   <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">s</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">s</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> </span><br/>

<a NAME="E2:50"/><span class="lab"><font color="Green" title="E41">A2</font></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> <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></span>;<br/>
<a NAME="E3:50"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> 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="50_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> or  not <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> or  not the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x2</font> or <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50_1"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:50_1"/><span class="lab"><font color="Green" title="E44">A5</font></span>: 
<font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:50_1"/><span class="lab"><font color="Green" title="E45">A6</font></span>: 
the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">X1</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><span class="kw"> such that </span><br/><a NAME="E5:50_1"/><span class="lab"><font color="Green" title="E46">A7</font></span>: 
<font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a></span>;<br/>
<a NAME="E6:50_1"/>
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a></span>;<br/>
<a NAME="E7:50_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A8</font></span>: 
 ex <font color="Olive" title="b1">X11</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">X11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">X11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E41">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">X2</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><span class="kw"> such that </span><br/><a NAME="E9:50_1"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
<font color="Maroon" title="c7">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a></span>;<br/>
<a NAME="E10:50_1"/>
<font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a></span>;<br/>
<a NAME="E11:50_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A10</font></span>: 
 ex <font color="Olive" title="b1">X12</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">X12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">X12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E41">A2</font></span></a></span>;<br/>
<a NAME="E12:50_1"/><span class="lab"><font color="Green" title="E50">A11</font></span>: 
<span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c7">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E9:50_1"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="ref" href="combgras.html#D14" target="_self" title="COMBGRAS:def.14">Def14</a></span>;<br/>
<a NAME="E13:50_1"/>
<span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E5:50_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="ref" href="combgras.html#D14" target="_self" title="COMBGRAS:def.14">Def14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:50_1"/>
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="txt" href="#E5:50_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E9:50_1"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="txt" href="#E7:50_1"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E11:50_1"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="txt" href="#E12:50_1"><span class="lab"><font color="Green" title="E50">A11</font></span></a>, <a class="ref" href="combgras.html#T6" target="_self" title="COMBGRAS:th.6">Th6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:50"/>
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> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <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="50_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">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> implies  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="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_2"/>
<font color="Maroon" title="c4">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )</span><br/>

<a NAME="E2:50_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A12</font></span>: 
 ex <font color="Olive" title="b1">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E41">A2</font></span></a></span>;<br/>
<a NAME="E3:50_2"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T167" title="RELAT_1:th.167">RELAT_1:167</a></span>;<br/>
<a NAME="E4:50_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A14</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></span>;<br/>
<a NAME="E5:50_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E6:50_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A15</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_2"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="ref" href="funct_1.html#T147" title="FUNCT_1:th.147">FUNCT_1:147</a></span>;<br/>
<a NAME="E7:50_2"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_2"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E3:50_2"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="ref" href="combgras.html#T4" target="_self" title="COMBGRAS:th.4">Th4</a></span>;<br/>
<a NAME="E8:50_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">s</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c4">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E4:50_2"><span class="lab"><font color="Green" title="E45">A14</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">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><span class="kw"> such that </span><br/><a NAME="E10:50_2"/><span class="lab"><font color="Green" title="E47">A16</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c4">y</font>
 ;<br/>
<a NAME="E11:50_2"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E6:50_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="txt" href="#E10:50_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="ref" href="combgras.html#D14" target="_self" title="COMBGRAS:def.14">Def14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:50_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="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:50"/><span class="kw">then </span>
 <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 <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T16" title="FUNCT_2:th.16">FUNCT_2:16</a></span>;<br/>
<a NAME="E6:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A17</font></span>: 
the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E7:50"/><span class="lab"><font color="Green" title="E44">A18</font></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> <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></span>;<br/>
<a NAME="E8:50"/>
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#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> 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#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> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <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="50_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">y</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> implies  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#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> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_3"/>
<font color="Maroon" title="c4">y</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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#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> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )</span><br/>

<a NAME="E2:50_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A19</font></span>: 
 ex <font color="Olive" title="b1">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E44">A18</font></span></a></span>;<br/>
<a NAME="E3:50_3"/><span class="lab"><font color="Green" title="E46">A20</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T167" title="RELAT_1:th.167">RELAT_1:167</a></span>;<br/>
<a NAME="E4:50_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A21</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></span>;<br/>
<a NAME="E5:50_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E6:50_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A22</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_3"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="ref" href="funct_1.html#T147" title="FUNCT_1:th.147">FUNCT_1:147</a></span>;<br/>
<a NAME="E7:50_3"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_3"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E3:50_3"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="ref" href="combgras.html#T4" target="_self" title="COMBGRAS:th.4">Th4</a></span>;<br/>
<a NAME="E8:50_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">s</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c4">y</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="txt" href="#E4:50_3"><span class="lab"><font color="Green" title="E47">A21</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">A</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E10:50_3"/><span class="lab"><font color="Green" title="E49">A23</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c4">y</font>
 ;<br/>
<a NAME="E11:50_3"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E6:50_3"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="txt" href="#E10:50_3"><span class="lab"><font color="Green" title="E49">A23</font></span></a>, <a class="ref" href="combgras.html#D14" target="_self" title="COMBGRAS:def.14">Def14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:50_3"/>
 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#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> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:50"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T16" title="FUNCT_2:th.16">FUNCT_2:16</a></span>;<br/>
<a NAME="E10:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A24</font></span>: 
the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E11:50"/><span class="lab"><font color="Green" title="E46">A25</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></span>;<br/>
<a NAME="E12:50"/><span class="lab"><font color="Green" title="E47">A26</font></span>: 
 <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font> is  <a href="combgras.html#V6" title="COMBGRAS:attr.6">incidence_preserving</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="combgras.html#D8" target="_self" title="COMBGRAS:def.8">COMBGRAS:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">L1</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> holds <br/> ( <font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">L1</font> iff <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Olive" title="b1">L1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">L1</font> iff <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c5">L1</font> )</span><br/>



<a NAME="E1:50_4"/><span class="lab"><font color="Green" title="E48">A27</font></span>: 
( <font color="Maroon" title="c3">s</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c4">A1</font> &amp; <font color="Maroon" title="c3">s</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c5">L1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="combgras.html#D14" target="_self" title="COMBGRAS:def.14">Def14</a></span>;<br/>
<a NAME="E2:50_4"/>
<font color="Maroon" title="c4">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/>
<a NAME="E3:50_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A28</font></span>: 
 ex <font color="Olive" title="b1">a1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E41">A2</font></span></a></span>;<br/>
<a NAME="E4:50_4"/><span class="lab"><font color="Green" title="E50">A29</font></span>: 
( <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c5">L1</font> implies <font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">L1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:50_4_1"/>
<span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c5">L1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">L1</font></span><br/>

<a NAME="E2:50_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">s</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">s</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E1:50_4"><span class="lab"><font color="Green" title="E48">A27</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E3:50_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:50"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="txt" href="#E3:50_4"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="funct_1.html#T157" title="FUNCT_1:th.157">FUNCT_1:157</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:50_4_1"/>
<font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:50_4"/>
( <font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">L1</font> implies <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c5">L1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:50_4_2"/>
<font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">L1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c5">L1</font></span><br/>

<a NAME="E2:50_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>;<br/>
<a NAME="E3:50_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">s</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">s</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:50_4_2"/>
<span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c5">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E1:50_4"><span class="lab"><font color="Green" title="E48">A27</font></span></a>, <a class="ref" href="combgras.html#T10" target="_self" title="COMBGRAS:th.10">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:50_4"/>
( <font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">L1</font> iff <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c5">L1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_4"><span class="lab"><font color="Green" title="E50">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:50"/>
the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> 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="50_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> or  not <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> or  not the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x2</font> or <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50_5"/><span class="lab"><font color="Green" title="E48">A30</font></span>: 
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:50_5"/><span class="lab"><font color="Green" title="E49">A31</font></span>: 
<font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:50_5"/><span class="lab"><font color="Green" title="E50">A32</font></span>: 
the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">X1</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:50_5"/><span class="lab"><font color="Green" title="E51">A33</font></span>: 
<font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_5"><span class="lab"><font color="Green" title="E48">A30</font></span></a></span>;<br/>
<a NAME="E6:50_5"/>
<font color="Maroon" title="c4">x1</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_5"><span class="lab"><font color="Green" title="E48">A30</font></span></a></span>;<br/>
<a NAME="E7:50_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A34</font></span>: 
 ex <font color="Olive" title="b1">X11</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">X11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">X11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E44">A18</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">X2</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E9:50_5"/><span class="lab"><font color="Green" title="E53">A35</font></span>: 
<font color="Maroon" title="c7">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_5"><span class="lab"><font color="Green" title="E49">A31</font></span></a></span>;<br/>
<a NAME="E10:50_5"/>
<font color="Maroon" title="c5">x2</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_5"><span class="lab"><font color="Green" title="E49">A31</font></span></a></span>;<br/>
<a NAME="E11:50_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A36</font></span>: 
 ex <font color="Olive" title="b1">X12</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st <br/>( <font color="Olive" title="b1">X12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">X12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E44">A18</font></span></a></span>;<br/>
<a NAME="E12:50_5"/><span class="lab"><font color="Green" title="E55">A37</font></span>: 
<span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c7">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E9:50_5"><span class="lab"><font color="Green" title="E53">A35</font></span></a>, <a class="ref" href="combgras.html#D14" target="_self" title="COMBGRAS:def.14">Def14</a></span>;<br/>
<a NAME="E13:50_5"/>
<span class="p1">(<span class="default"><a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E5:50_5"><span class="lab"><font color="Green" title="E51">A33</font></span></a>, <a class="ref" href="combgras.html#D14" target="_self" title="COMBGRAS:def.14">Def14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:50_5"/>
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_5"><span class="lab"><font color="Green" title="E50">A32</font></span></a>, <a class="txt" href="#E5:50_5"><span class="lab"><font color="Green" title="E51">A33</font></span></a>, <a class="txt" href="#E9:50_5"><span class="lab"><font color="Green" title="E53">A35</font></span></a>, <a class="txt" href="#E7:50_5"><span class="lab"><font color="Green" title="E52">A34</font></span></a>, <a class="txt" href="#E11:50_5"><span class="lab"><font color="Green" title="E54">A36</font></span></a>, <a class="txt" href="#E12:50_5"><span class="lab"><font color="Green" title="E55">A37</font></span></a>, <a class="ref" href="combgras.html#T6" target="_self" title="COMBGRAS:th.6">Th6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E14:50"/>
 <a href="combgras.html#K8" title="COMBGRAS:func.8">incprojmap</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c3">s</font> is  <a href="combgras.html#V7" title="COMBGRAS:attr.7">automorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:50"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="txt" href="#E6:50"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="txt" href="#E12:50"><span class="lab"><font color="Green" title="E47">A26</font></span></a>, <a class="ref" href="combgras.html#D9" target="_self" title="COMBGRAS:def.9">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
