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

<b>let </b><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font> be    <a href="incsp_1.html#L1" title="INCSP_1:struct.1">IncProjStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> being   <a href="combgras.html#L1" title="COMBGRAS:struct.1">IncProjMap</a> of <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font>  st ( for <font color="Olive" title="b3">A</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S1</font> holds  <font color="Olive" title="b1">F1</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F2</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Olive" title="b3">A</font> ) &amp; ( for <font color="Olive" title="b3">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S1</font> holds  <font color="Olive" title="b1">F1</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Olive" title="b3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F2</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Olive" title="b3">L</font> ) holds <br/> <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(# the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b1">F1</font>,the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b1">F1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(# the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Olive" title="b2">F2</font>,the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Olive" title="b2">F2</font> #)</span><br/><b>let </b><font color="Maroon" title="c3">F1</font>, <font color="Maroon" title="c4">F2</font> be    <a href="combgras.html#L1" title="COMBGRAS:struct.1">IncProjMap</a> of <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</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 <font color="Maroon" title="c1">S1</font> holds  <font color="Maroon" title="c3">F1</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="c4">F2</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</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 <font color="Maroon" title="c1">S1</font> holds  <font color="Maroon" title="c3">F1</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="c4">F2</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Olive" title="b1">L</font> ) implies  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(# the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F1</font>,the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(# the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c4">F2</font>,the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c4">F2</font> #) )</span><br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E21">A1</font></i>: 
( ( for <font color="Olive" title="b1">A</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S1</font> holds  <font color="Maroon" title="c3">F1</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="c4">F2</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</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 <font color="Maroon" title="c1">S1</font> holds  <font color="Maroon" title="c3">F1</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="c4">F2</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Olive" title="b1">L</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(# the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F1</font>,the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(# the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c4">F2</font>,the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c4">F2</font> #)</span><br/>

<a NAME="E2:25"/><i><font color="Green" title="E22">A2</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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 <font color="Maroon" title="c1">S1</font> holds <br/>the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</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 <font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">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 <font color="Maroon" title="c1">S1</font> implies the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</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 <font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> )</span><br/>

<b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon" title="c5">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 <font color="Maroon" title="c1">S1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</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 <font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span><br/>

<b>then consider </b><font color="Maroon" title="c6">A</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S1</font><b> such that </b><br/><a NAME="E3:25_1"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 ;<br/>
<a NAME="E4:25_1"/>
<font color="Maroon" title="c3">F1</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F2</font> <a href="combgras.html#K2" title="COMBGRAS:func.2">.</a> <font color="Maroon" title="c6">A</font>
 
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:25_1"/>
the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</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 <font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="#E3:25_1"><i><font color="Green" title="E23">A3</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>
<a NAME="E3:25"/>
for <font color="Olive" title="b1">l</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">l</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 <font color="Maroon" title="c1">S1</font> holds <br/>the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">l</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 <font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">l</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">l</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 <font color="Maroon" title="c1">S1</font> implies the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">l</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 <font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">l</font> )</span><br/>

<b>assume </b><a NAME="E1:25_2"/>
<font color="Maroon" title="c5">l</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 <font color="Maroon" title="c1">S1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">l</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 <font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">l</font></span><br/>

<b>then consider </b><font color="Maroon" title="c6">L</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S1</font><b> such that </b><br/><a NAME="E3:25_2"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font>
 ;<br/>
<a NAME="E4:25_2"/>
<font color="Maroon" title="c3">F1</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F2</font> <a href="combgras.html#K3" title="COMBGRAS:func.3">.</a> <font color="Maroon" title="c6">L</font>
 
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:25_2"/>
the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">l</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 <font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">l</font>
 <b>by </b><i><a class="txt" href="#E3:25_2"><i><font color="Green" title="E23">A4</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>
<a NAME="E4:25"/><b>then </b>
( the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c4">F2</font> &amp; the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c4">F2</font> )
 
<b>by </b><i><a class="txt" href="#E2:25"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/>
<b>hence </b><a NAME="E5:25"/>
 <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(# the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c3">F1</font>,the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c3">F1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="combgras.html#G1" title="COMBGRAS:aggr.1">IncProjMap</a>(# the <a href="combgras.html#U1" title="COMBGRAS:sel.1">point-map</a> of <font color="Maroon" title="c4">F2</font>,the <a href="combgras.html#U2" title="COMBGRAS:sel.2">line-map</a> of <font color="Maroon" title="c4">F2</font> #)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
