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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c3">s</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> $1;<br/>
<a NAME="E1:39_1_1"/><i><font color="Green" title="E30">A2</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#E1:39"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></i>;<br/>
<a NAME="E2:39_1_1"/><i><font color="Green" title="E31">A3</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#E1:39"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="combgras.html#D1" target="_self" title="COMBGRAS:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">P</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:39_1_1"/><i><font color="Green" title="E32">A4</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">P</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> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<b>consider </b><font color="Maroon" title="c5">L</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:39_1_1"/><i><font color="Green" title="E33">A5</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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/><font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E7:39_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">P</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="39_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">b</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">b</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">P</font> or <font color="Maroon" title="c6">b</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><br/>

<b>assume </b><a NAME="E1:39_1_1_1"/>
<font color="Maroon" title="c6">b</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">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">b</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><br/>

<b>then consider </b><font color="Maroon" title="c7">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_1"/><i><font color="Green" title="E34">A6</font></i>: 
( <font color="Maroon" title="c7">a</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="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> )
 <b>by </b><i><a class="txt" href="combgras.html#E4:39_1_1"><i><font color="Green" title="E32">A4</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="c8">A</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="E5:39_1_1_1"/><i><font color="Green" title="E35">A7</font></i>: 
( <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c8">A</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#E1:39_1_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="combgras.html#E3:39_1_1_1"><i><font color="Green" title="E34">A6</font></i></a></i>;<br/>
<a NAME="E6:39_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:39_1_1_1"/><b>then </b><i><font color="Green" title="E36">A8</font></i>: 
( <font color="Maroon" title="c6">b</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="c8">A</font> &amp; <font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">s</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E4:39_1_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="combgras.html#E3:39_1_1_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="combgras.html#E5:39_1_1_1"><i><font color="Green" title="E35">A7</font></i></a></i>;<br/>
<a NAME="E8:39_1_1_1"/><b>then </b><i><font color="Green" title="E37">A9</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">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#E5:39_1_1_1"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="combgras.html#T4" target="_self" title="COMBGRAS:th.4">Th4</a></i>;<br/>
<a NAME="E9:39_1_1_1"/>
<font color="Maroon" title="c6">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b</font> or <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><br/>

<b>assume </b><a NAME="E1:39_1_1_1_1"/><i><font color="Green" title="E38">A10</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/>

<b>consider </b><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_1_1"/><i><font color="Green" title="E39">A11</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> )
 <b>by </b><i><a class="txt" href="combgras.html#E7:39_1_1_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="txt" href="combgras.html#E1:39_1_1_1_1"><i><font color="Green" title="E38">A10</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="E4:39_1_1_1_1"/>
( <font color="Maroon" title="c9">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="c3">s</font> &amp;  <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> )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:39_1_1_1_1"><i><font color="Green" title="E39">A11</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <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="E5:39_1_1_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:39_1_1_1"/>
<font color="Maroon" title="c6">b</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#E1:39_1_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="combgras.html#E8:39_1_1_1"><i><font color="Green" title="E37">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">P</font> = <font color="Maroon" title="c4">P</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#E4:39_1_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E9:39_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">b</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">b</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="c5">L</font> or <font color="Maroon" title="c7">b</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><br/>

<b>assume </b><a NAME="E1:39_1_1_2"/>
<font color="Maroon" title="c7">b</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="c5">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">b</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><br/>

<b>then consider </b><font color="Maroon" title="c8">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_2"/><i><font color="Green" title="E34">A12</font></i>: 
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="combgras.html#K1" title="COMBGRAS:func.1">G_</a> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> )
 <b>by </b><i><a class="txt" href="combgras.html#E6:39_1_1"><i><font color="Green" title="E33">A5</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="c9">A</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="E5:39_1_1_2"/><i><font color="Green" title="E35">A13</font></i>: 
( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c9">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 )
 <b>by </b><i><a class="txt" href="combgras.html#E2:39_1_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="combgras.html#E3:39_1_1_2"><i><font color="Green" title="E34">A12</font></i></a></i>;<br/>
<a NAME="E6:39_1_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:39_1_1_2"/><b>then </b><i><font color="Green" title="E36">A14</font></i>: 
( <font color="Maroon" title="c7">b</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="c9">A</font> &amp; <font color="Maroon" title="c9">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">s</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E6:39_1_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="combgras.html#E3:39_1_1_2"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="combgras.html#E5:39_1_1_2"><i><font color="Green" title="E35">A13</font></i></a></i>;<br/>
<a NAME="E8:39_1_1_2"/><b>then </b><i><font color="Green" title="E37">A15</font></i>: 
 <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> <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#E5:39_1_1_2"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="combgras.html#T4" target="_self" title="COMBGRAS:th.4">Th4</a></i>;<br/>
<a NAME="E9:39_1_1_2"/>
<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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">b</font> or <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><br/>

<b>assume </b><a NAME="E1:39_1_1_2_1"/><i><font color="Green" title="E38">A16</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/>

<b>consider </b><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_2_1"/><i><font color="Green" title="E39">A17</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> )
 <b>by </b><i><a class="txt" href="combgras.html#E7:39_1_1_2"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="combgras.html#E1:39_1_1_2_1"><i><font color="Green" title="E38">A16</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="E4:39_1_1_2_1"/>
( <font color="Maroon" title="c10">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="c3">s</font> &amp;  <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> )
 
<b>by </b><i><a class="txt" href="combgras.html#E3:39_1_1_2_1"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <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="E5:39_1_1_2_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:39_1_1_2"/>
<font color="Maroon" title="c7">b</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>
 <b>by </b><i><a class="txt" href="combgras.html#E2:39_1_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="combgras.html#E8:39_1_1_2"><i><font color="Green" title="E37">A15</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">L</font> = <font color="Maroon" title="c5">L</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#E6:39_1_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<b>take </b><font color="Maroon" title="c8">F</font> =  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(# <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">L</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">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> holds  <font color="Maroon" title="c8">F</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> <font color="Maroon" title="c3">s</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">A</font> ) &amp; ( for <font color="Olive" title="b1">L</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  <font color="Maroon" title="c8">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Olive" title="b1">L</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="Olive" title="b1">L</font> ) )</span><br/>

<b>thus </b><a NAME="E11:39_1_1"/>
( ( 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> holds  <font color="Maroon" title="c8">F</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> <font color="Maroon" title="c3">s</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">A</font> ) &amp; ( for <font color="Olive" title="b1">L</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  <font color="Maroon" title="c8">F</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Olive" title="b1">L</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="Olive" title="b1">L</font> ) )
 <b>by </b><i><a class="txt" href="combgras.html#E4:39_1_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="combgras.html#E6:39_1_1"><i><font color="Green" title="E33">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
