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

<b>let </b><font color="Maroon" title="c1">IPP</font> be  <a href="incproj.html#V5" title="INCPROJ:attr.5">2-dimensional</a> <a href="incproj.html#V9" title="INCPROJ:attr.9">Desarguesian</a> <a href="incproj.html#NM2" title="INCPROJ:NM.2">IncProjSp</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">c</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>;<br/><b>let </b><font color="Maroon" title="c5">K</font>, <font color="Maroon" title="c6">L</font>, <font color="Maroon" title="c7">R</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font>;<br/>













<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E20">A1</font></i>: 
( not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font> &amp; not <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font> &amp; not <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <font color="Maroon" title="c5">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">R</font> )
 ;<br/>

<a NAME="E2:28"/><i><font color="Green" title="E21">A2</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <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 <font color="Maroon" title="c1">IPP</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <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 <font color="Maroon" title="c1">IPP</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c8">f</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>;<br/>
<a NAME="E1:28_1"/><i><font color="Green" title="E22">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></i>;<br/>
<a NAME="E2:28_1"/><i><font color="Green" title="E23">A4</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></i>;<br/>
<b>thus </b><a NAME="E3:28_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <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 <font color="Maroon" title="c1">IPP</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_1"><i><font color="Green" title="E22">A3</font></i></a></i>;<br/>

<b>thus </b><a NAME="E4:28_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <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 <font color="Maroon" title="c1">IPP</font>
 <b>by </b><i><a class="txt" href="projred1.html#E2:28_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">k</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> );<br/>
<b>consider </b><font color="Maroon" title="c8">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:28"/><i><font color="Green" title="E22">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> iff ( <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 <font color="Maroon" title="c1">IPP</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<div><i><font color="Green" title="E23">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_2"/><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font>
 ;<br/><a NAME="E2:28_2"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_2_1"/><i><font color="Green" title="E25">A8</font></i>: 
( <font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:28_2_1_1"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_2_1_1_1"/><i><font color="Green" title="E26">A9</font></i>: 
<font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_2_1_1_1_1"/><i><font color="Green" title="E27">A10</font></i>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">e</font> = <font color="Maroon" title="c9">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<a NAME="E3:28_2_1_1_1_1"/><i><font color="Green" title="E28">A11</font></i>: 
 ex <font color="Olive" title="b1">e'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e'</font> &amp; <font color="Olive" title="b1">e'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="projred1.html#E1:28_2_1_1_1_1"><i><font color="Green" title="E27">A10</font></i></a></i>;<br/>
<a NAME="E4:28_2_1_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></i>;<br/>
<b>hence </b><a NAME="E5:28_2_1_1_1_1"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E3:28_2_1_1_1_1"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_2_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_2_1_1_1_2"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:28_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E27">A12</font></i>: 
<font color="Maroon" title="c9">e</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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">e</font> = <font color="Maroon" title="c9">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> ;<br/>
<a NAME="E4:28_2_1_1_1_2"/>
<font color="Maroon" title="c10">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E2:28_2_1_1_1_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:28_2_1_1_1_2"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_2_1_1_1"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_2_1_1_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:28_2_1_1"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:28_2_1_1_2"/><i><font color="Green" title="E26">A13</font></i>: 
<font color="Maroon" title="c8">X</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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_2_1_1_2_1"/><i><font color="Green" title="E27">A14</font></i>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">e</font> = <font color="Maroon" title="c9">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<a NAME="E3:28_2_1_1_2_1"/>
 ex <font color="Olive" title="b1">e'</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e'</font> &amp; <font color="Olive" title="b1">e'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="projred1.html#E1:28_2_1_1_2_1"><i><font color="Green" title="E27">A14</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:28_2_1_1_2_1"/>
<font color="Maroon" title="c9">e</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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_2_1_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_2_1_1_2_2"/><i><font color="Green" title="E27">A15</font></i>: 
<font color="Maroon" title="c9">e</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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">e</font> = <font color="Maroon" title="c9">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> ;<br/>
<a NAME="E3:28_2_1_1_2_2"/>
<font color="Maroon" title="c10">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="projred1.html#E1:28_2_1_1_2_2"><i><font color="Green" title="E27">A15</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:28_2_1_1_2_2"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_2_1_1_2"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_2_1_1_2"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E2:28_2_1"/><i><font color="Green" title="E26">A16</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>;<br/>

<b>assume </b><a NAME="E1:28_2_1_2"/><i><font color="Green" title="E27">A17</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ;<br/>

<a NAME="E2:28_2_1_2"/><b>then </b><i><font color="Green" title="E28">A18</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="projred1.html#E1:28_2_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<a NAME="E3:28_2_1_2"/><i><font color="Green" title="E29">A19</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="projred1.html#E1:28_2_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="projred1.html#E1:28_2_1_2"><i><font color="Green" title="E27">A17</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">y</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E2:28_2_1_2"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="projred1.html#T22" target="_self" title="PROJRED1:th.22">Th22</a></i>;<br/>
<a NAME="E5:28_2_1_2"/><i><font color="Green" title="E30">A20</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E2:28_2_1_2"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">A</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E7:28_2_1_2"/><i><font color="Green" title="E31">A21</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c10">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">B</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E9:28_2_1_2"/><i><font color="Green" title="E32">A22</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">B</font> &amp; <font color="Maroon" title="c9">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">B</font> &amp; <font color="Maroon" title="c10">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">B</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E2:28_2_1_2"><i><font color="Green" title="E28">A18</font></i></a>, <a class="txt" href="projred1.html#E5:28_2_1_2"><i><font color="Green" title="E30">A20</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">z</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="projred1.html#E5:28_2_1_2"><i><font color="Green" title="E30">A20</font></i></a>, <a class="ref" href="projred1.html#T22" target="_self" title="PROJRED1:th.22">Th22</a></i>;<br/>
<a NAME="E11:28_2_1_2"/><i><font color="Green" title="E33">A23</font></i>: 
<font color="Maroon" title="c13">z</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="projred1.html#E5:28_2_1_2"><i><font color="Green" title="E30">A20</font></i></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">C</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E13:28_2_1_2"/><i><font color="Green" title="E34">A24</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">C</font> &amp; <font color="Maroon" title="c10">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">C</font> &amp; <font color="Maroon" title="c13">z</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">C</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="projred1.html#E5:28_2_1_2"><i><font color="Green" title="E30">A20</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<a NAME="E14:28_2_1_2"/><i><font color="Green" title="E35">A25</font></i>: 
<font color="Maroon" title="c11">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">B</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E5:28_2_1_2"><i><font color="Green" title="E30">A20</font></i></a>, <a class="txt" href="projred1.html#E7:28_2_1_2"><i><font color="Green" title="E31">A21</font></i></a>, <a class="txt" href="projred1.html#E9:28_2_1_2"><i><font color="Green" title="E32">A22</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E15:28_2_1_2"/>
<font color="Maroon" title="c11">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">C</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E5:28_2_1_2"><i><font color="Green" title="E30">A20</font></i></a>, <a class="txt" href="projred1.html#E7:28_2_1_2"><i><font color="Green" title="E31">A21</font></i></a>, <a class="txt" href="projred1.html#E13:28_2_1_2"><i><font color="Green" title="E34">A24</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E16:28_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="projred1.html#E2:28_2_1_2"><i><font color="Green" title="E28">A18</font></i></a>, <a class="txt" href="projred1.html#E3:28_2_1_2"><i><font color="Green" title="E29">A19</font></i></a>, <a class="txt" href="projred1.html#E9:28_2_1_2"><i><font color="Green" title="E32">A22</font></i></a>, <a class="txt" href="projred1.html#E11:28_2_1_2"><i><font color="Green" title="E33">A23</font></i></a>, <a class="txt" href="projred1.html#E13:28_2_1_2"><i><font color="Green" title="E34">A24</font></i></a>, <a class="txt" href="projred1.html#E14:28_2_1_2"><i><font color="Green" title="E35">A25</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:28_2_1"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>,the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:28"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>hence </b><a NAME="E4:28_2_1"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_2_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="projred1.html#E2:28_2_1"><i><font color="Green" title="E26">A16</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_2"/>
 ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_2"><i><font color="Green" title="E24">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3"/><i><font color="Green" title="E24">A26</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font>
 ;<br/><b>consider </b><font color="Maroon" title="c9">A</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E3:28_3"/><i><font color="Green" title="E25">A27</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">c1</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E5:28_3"/><i><font color="Green" title="E26">A28</font></i>: 
( <font color="Maroon" title="c10">c1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; <font color="Maroon" title="c10">c1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14" title="INCPROJ:def.14">INCPROJ:def 14</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">a1</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E7:28_3"/><i><font color="Green" title="E27">A29</font></i>: 
( <font color="Maroon" title="c11">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; <font color="Maroon" title="c10">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">a1</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">a1</font> )
 <b>by </b><i><a class="ref" href="projred1.html#T8" target="_self" title="PROJRED1:th.8">Th8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">a2</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a1</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="ref" href="projred1.html#T22" target="_self" title="PROJRED1:th.22">Th22</a></i>;<br/><a NAME="E9:28_3"/><i><font color="Green" title="E28">A30</font></i>: 
<font color="Maroon" title="c12">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">a3</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a2</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T22" target="_self" title="PROJRED1:th.22">Th22</a></i>;<br/><a NAME="E11:28_3"/><i><font color="Green" title="E29">A31</font></i>: 
<font color="Maroon" title="c13">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E9:28_3"><i><font color="Green" title="E28">A30</font></i></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">B</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E13:28_3"/><i><font color="Green" title="E30">A32</font></i>: 
( <font color="Maroon" title="c11">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">B</font> &amp; <font color="Maroon" title="c13">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">B</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">o</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E15:28_3"/><i><font color="Green" title="E31">A33</font></i>: 
( <font color="Maroon" title="c15">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c15">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">B</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14" title="INCPROJ:def.14">INCPROJ:def 14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">c2</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">c1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E5:28_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="ref" href="projred1.html#T22" target="_self" title="PROJRED1:th.22">Th22</a></i>;<br/><a NAME="E17:28_3"/><i><font color="Green" title="E32">A34</font></i>: 
<font color="Maroon" title="c16">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E5:28_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">c3</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">c2</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T22" target="_self" title="PROJRED1:th.22">Th22</a></i>;<br/><a NAME="E19:28_3"/><i><font color="Green" title="E33">A35</font></i>: 
<font color="Maroon" title="c17">c3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E17:28_3"><i><font color="Green" title="E32">A34</font></i></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></i>;<br/><a NAME="E20:28_3"/><i><font color="Green" title="E34">A36</font></i>: 
not <font color="Maroon" title="c11">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/><a NAME="E21:28_3"/><i><font color="Green" title="E35">A37</font></i>: 
not <font color="Maroon" title="c13">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_3_1"/>
<font color="Maroon" title="c13">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 ;<br/>

<a NAME="E2:28_3_1"/><b>then </b><i><font color="Green" title="E36">A38</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a3</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E11:28_3"><i><font color="Green" title="E29">A31</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E3:28_3_1"/>
 ex <font color="Olive" title="b1">C</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">C</font> )
 
<b>by </b><i><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/>
<a NAME="E4:28_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a3</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_1"><i><font color="Green" title="E36">A38</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<a NAME="E5:28_3_1"/><b>then </b><i><font color="Green" title="E37">A39</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a2</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E9:28_3"><i><font color="Green" title="E28">A30</font></i></a>, <a class="ref" href="projred1.html#T26" target="_self" title="PROJRED1:th.26">Th26</a></i>;<br/>
<a NAME="E6:28_3_1"/>
 ex <font color="Olive" title="b1">D</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">D</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">D</font> )
 
<b>by </b><i><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/>
<a NAME="E7:28_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a2</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E5:28_3_1"><i><font color="Green" title="E37">A39</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E8:28_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="ref" href="projred1.html#T26" target="_self" title="PROJRED1:th.26">Th26</a></i>;<br/>


</div><b>end;</b></div><a NAME="E22:28_3"/><i><font color="Green" title="E36">A40</font></i>: 
( not <font color="Maroon" title="c15">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Maroon" title="c15">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_3_2"/><i><font color="Green" title="E37">A41</font></i>: 
( <font color="Maroon" title="c15">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> or <font color="Maroon" title="c15">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> )
 ;<br/>

<div><i><font color="Green" title="E38">A42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3_2_1"/><i><font color="Green" title="E39">A43</font></i>: 
<font color="Maroon" title="c15">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 ;<br/><a NAME="E2:28_3_2_1"/><b>then </b>
<font color="Maroon" title="c15">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c1</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E3:28_3"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="projred1.html#E5:28_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="projred1.html#E15:28_3"><i><font color="Green" title="E31">A33</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/><b>hence </b><a NAME="E3:28_3_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E7:28_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="projred1.html#E13:28_3"><i><font color="Green" title="E30">A32</font></i></a>, <a class="txt" href="projred1.html#E15:28_3"><i><font color="Green" title="E31">A33</font></i></a>, <a class="txt" href="projred1.html#E21:28_3"><i><font color="Green" title="E35">A37</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_2_1"><i><font color="Green" title="E39">A43</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3_2_2"/><i><font color="Green" title="E39">A44</font></i>: 
<font color="Maroon" title="c15">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font>
 ;<br/><a NAME="E2:28_3_2_2"/><i><font color="Green" title="E40">A45</font></i>: 
 ex <font color="Olive" title="b1">A1</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A1</font> &amp; <font color="Maroon" title="c10">c1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A1</font> &amp; <font color="Maroon" title="c16">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A1</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E5:28_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="projred1.html#E17:28_3"><i><font color="Green" title="E32">A34</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c18">A2</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E4:28_3_2_2"/><i><font color="Green" title="E41">A46</font></i>: 
( <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">A2</font> &amp; <font color="Maroon" title="c16">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">A2</font> &amp; <font color="Maroon" title="c17">c3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">A2</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E17:28_3"><i><font color="Green" title="E32">A34</font></i></a>, <a class="txt" href="projred1.html#E19:28_3"><i><font color="Green" title="E33">A35</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/><a NAME="E5:28_3_2_2"/>
<font color="Maroon" title="c16">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E3:28_3"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="projred1.html#E5:28_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_2_2"><i><font color="Green" title="E40">A45</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/><a NAME="E6:28_3_2_2"/><b>then </b>
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">A2</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E3:28_3"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="projred1.html#E17:28_3"><i><font color="Green" title="E32">A34</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_2_2"><i><font color="Green" title="E41">A46</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/><a NAME="E7:28_3_2_2"/><b>then </b><i><font color="Green" title="E42">A47</font></i>: 
<font color="Maroon" title="c15">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">c3</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E15:28_3"><i><font color="Green" title="E31">A33</font></i></a>, <a class="txt" href="projred1.html#E19:28_3"><i><font color="Green" title="E33">A35</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_2_2"><i><font color="Green" title="E39">A44</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_2_2"><i><font color="Green" title="E41">A46</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/><a NAME="E8:28_3_2_2"/>
<font color="Maroon" title="c15">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a3</font>
 <b>by </b><i><a class="txt" href="projred1.html#E11:28_3"><i><font color="Green" title="E29">A31</font></i></a>, <a class="txt" href="projred1.html#E13:28_3"><i><font color="Green" title="E30">A32</font></i></a>, <a class="txt" href="projred1.html#E15:28_3"><i><font color="Green" title="E31">A33</font></i></a>, <a class="txt" href="projred1.html#E20:28_3"><i><font color="Green" title="E34">A36</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_2_2"><i><font color="Green" title="E39">A44</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/><a NAME="E9:28_3_2_2"/><b>then </b>
<font color="Maroon" title="c16">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a2</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E9:28_3"><i><font color="Green" title="E28">A30</font></i></a>, <a class="txt" href="projred1.html#E17:28_3"><i><font color="Green" title="E32">A34</font></i></a>, <a class="txt" href="projred1.html#E7:28_3_2_2"><i><font color="Green" title="E42">A47</font></i></a>, <a class="ref" href="projred1.html#T26" target="_self" title="PROJRED1:th.26">Th26</a></i>;<br/><b>hence </b><a NAME="E10:28_3_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E5:28_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="ref" href="projred1.html#T26" target="_self" title="PROJRED1:th.26">Th26</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:28_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_2"><i><font color="Green" title="E37">A41</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_2"><i><font color="Green" title="E38">A42</font></i></a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green" title="E37">A48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3_3"/><i><font color="Green" title="E38">A49</font></i>: 
<font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font>
 ;<br/><b>take </b><font color="Maroon" title="c18">q</font> = <font color="Maroon" title="c3">q</font>;<br/><a NAME="E2:28_3_3"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_3_1"/><i><font color="Green" title="E39">A50</font></i>: 
( <font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:28_3_3_1_1"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_3_1_1_1"/><i><font color="Green" title="E40">A51</font></i>: 
<font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_3_1_1_1_1"/><i><font color="Green" title="E41">A52</font></i>: 
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c20">e</font> = <font color="Maroon" title="c19">e</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<a NAME="E3:28_3_3_1_1_1_1"/><i><font color="Green" title="E42">A53</font></i>: 
 ex <font color="Olive" title="b1">e'</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c20">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e'</font> &amp; <font color="Olive" title="b1">e'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3_1_1_1_1"><i><font color="Green" title="E41">A52</font></i></a></i>;<br/>
<a NAME="E4:28_3_3_1_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></i>;<br/>
<b>hence </b><a NAME="E5:28_3_3_1_1_1_1"/>
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E3:28_3_3_1_1_1_1"><i><font color="Green" title="E42">A53</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_3_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_3_1_1_1_2"/>
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:28_3_3_1_1_1_2"/><b>then </b><i><font color="Green" title="E41">A54</font></i>: 
<font color="Maroon" title="c19">e</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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">e</font> = <font color="Maroon" title="c19">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> ;<br/>
<a NAME="E4:28_3_3_1_1_1_2"/>
<font color="Maroon" title="c20">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_3_1_1_1_2"><i><font color="Green" title="E41">A54</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:28_3_3_1_1_1_2"/>
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_3_3_1_1_1"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_3_1_1_1"><i><font color="Green" title="E40">A51</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:28_3_3_1_1"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_3_1_1_2"/><i><font color="Green" title="E40">A55</font></i>: 
<font color="Maroon" title="c8">X</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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_3_1_1_2_1"/><i><font color="Green" title="E41">A56</font></i>: 
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c20">e</font> = <font color="Maroon" title="c19">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<a NAME="E3:28_3_3_1_1_2_1"/>
 ex <font color="Olive" title="b1">e'</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c20">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e'</font> &amp; <font color="Olive" title="b1">e'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3_1_1_2_1"><i><font color="Green" title="E41">A56</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_3_1_1_2_1"/>
<font color="Maroon" title="c19">e</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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3"><i><font color="Green" title="E38">A49</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_3_1_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_3_1_1_2_2"/><i><font color="Green" title="E41">A57</font></i>: 
<font color="Maroon" title="c19">e</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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c20">e</font> = <font color="Maroon" title="c19">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> ;<br/>
<a NAME="E3:28_3_3_1_1_2_2"/>
<font color="Maroon" title="c20">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3"><i><font color="Green" title="E38">A49</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3_1_1_2_2"><i><font color="Green" title="E41">A57</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_3_1_1_2_2"/>
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_3_3_1_1_2"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_3_1_1_2"><i><font color="Green" title="E40">A55</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E2:28_3_3_1"/><i><font color="Green" title="E40">A58</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>;<br/>

<b>assume </b><a NAME="E1:28_3_3_1_2"/><i><font color="Green" title="E41">A59</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ;<br/>

<a NAME="E2:28_3_3_1_2"/><b>then </b><i><font color="Green" title="E42">A60</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3"><i><font color="Green" title="E38">A49</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3_1"><i><font color="Green" title="E39">A50</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<a NAME="E3:28_3_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28_3_3"><i><font color="Green" title="E38">A49</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3_1"><i><font color="Green" title="E39">A50</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3_1_2"><i><font color="Green" title="E41">A59</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3"><i><font color="Green" title="E38">A49</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_3_1_2"><i><font color="Green" title="E42">A60</font></i></a>, <a class="ref" href="projred1.html#T21" target="_self" title="PROJRED1:th.21">Th21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:28_3_3_1"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>,the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:28"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_3_1"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_3_1"><i><font color="Green" title="E39">A50</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_3_1"><i><font color="Green" title="E40">A58</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_3_3"/>
 ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3"><i><font color="Green" title="E38">A49</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E38">A61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3_4"/><i><font color="Green" title="E39">A62</font></i>: 
<font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">R</font>
 ;<br/><b>take </b><font color="Maroon" title="c18">p</font> = <font color="Maroon" title="c2">p</font>;<br/><a NAME="E2:28_3_4"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c7">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_4_1"/><i><font color="Green" title="E40">A63</font></i>: 
( <font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:28_3_4_1_1"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_4_1_1_1"/><i><font color="Green" title="E41">A64</font></i>: 
<font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_4_1_1_1_1"/><i><font color="Green" title="E42">A65</font></i>: 
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c20">e</font> = <font color="Maroon" title="c19">e</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<a NAME="E3:28_3_4_1_1_1_1"/><i><font color="Green" title="E43">A66</font></i>: 
 ex <font color="Olive" title="b1">e'</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c20">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e'</font> &amp; <font color="Olive" title="b1">e'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4_1_1_1_1"><i><font color="Green" title="E42">A65</font></i></a></i>;<br/>
<a NAME="E4:28_3_4_1_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></i>;<br/>
<b>hence </b><a NAME="E5:28_3_4_1_1_1_1"/>
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E3:28_3_4_1_1_1_1"><i><font color="Green" title="E43">A66</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_4_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_4_1_1_1_2"/>
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:28_3_4_1_1_1_2"/><b>then </b><i><font color="Green" title="E42">A67</font></i>: 
<font color="Maroon" title="c19">e</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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">e</font> = <font color="Maroon" title="c19">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> ;<br/>
<a NAME="E4:28_3_4_1_1_1_2"/>
<font color="Maroon" title="c20">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_4_1_1_1_2"><i><font color="Green" title="E42">A67</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:28_3_4_1_1_1_2"/>
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_3_4_1_1_1"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_4_1_1_1"><i><font color="Green" title="E41">A64</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:28_3_4_1_1"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_4_1_1_2"/><i><font color="Green" title="E41">A68</font></i>: 
<font color="Maroon" title="c8">X</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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_4_1_1_2_1"/><i><font color="Green" title="E42">A69</font></i>: 
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c20">e</font> = <font color="Maroon" title="c19">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<a NAME="E3:28_3_4_1_1_2_1"/>
 ex <font color="Olive" title="b1">e'</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c20">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e'</font> &amp; <font color="Olive" title="b1">e'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4_1_1_2_1"><i><font color="Green" title="E42">A69</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_4_1_1_2_1"/>
<font color="Maroon" title="c19">e</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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4"><i><font color="Green" title="E39">A62</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_4_1_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_4_1_1_2_2"/><i><font color="Green" title="E42">A70</font></i>: 
<font color="Maroon" title="c19">e</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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c20">e</font> = <font color="Maroon" title="c19">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> ;<br/>
<a NAME="E3:28_3_4_1_1_2_2"/>
<font color="Maroon" title="c20">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4"><i><font color="Green" title="E39">A62</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4_1_1_2_2"><i><font color="Green" title="E42">A70</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_4_1_1_2_2"/>
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_3_4_1_1_2"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_4_1_1_2"><i><font color="Green" title="E41">A68</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E2:28_3_4_1"/><i><font color="Green" title="E41">A71</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>;<br/>

<b>assume </b><a NAME="E1:28_3_4_1_2"/><i><font color="Green" title="E42">A72</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ;<br/>

<a NAME="E2:28_3_4_1_2"/><b>then </b><i><font color="Green" title="E43">A73</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4"><i><font color="Green" title="E39">A62</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4_1"><i><font color="Green" title="E40">A63</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<a NAME="E3:28_3_4_1_2"/><i><font color="Green" title="E44">A74</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c7">R</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28_3_4"><i><font color="Green" title="E39">A62</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4_1"><i><font color="Green" title="E40">A63</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4_1_2"><i><font color="Green" title="E42">A72</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">y</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4"><i><font color="Green" title="E39">A62</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_4_1_2"><i><font color="Green" title="E43">A73</font></i></a>, <a class="ref" href="projred1.html#T22" target="_self" title="PROJRED1:th.22">Th22</a></i>;<br/>
<a NAME="E5:28_3_4_1_2"/>
<font color="Maroon" title="c20">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4"><i><font color="Green" title="E39">A62</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_4_1_2"><i><font color="Green" title="E43">A73</font></i></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></i>;<br/>
<b>hence </b><a NAME="E6:28_3_4_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4"><i><font color="Green" title="E39">A62</font></i></a>, <a class="txt" href="projred1.html#E3:28_3_4_1_2"><i><font color="Green" title="E44">A74</font></i></a>, <a class="ref" href="projred1.html#T21" target="_self" title="PROJRED1:th.21">Th21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:28_3_4_1"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>,the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:28"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_4_1"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c7">R</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_4_1"><i><font color="Green" title="E40">A63</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_4_1"><i><font color="Green" title="E41">A71</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_3_4"/>
 ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4"><i><font color="Green" title="E39">A62</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3_5"/><i><font color="Green" title="E39">A75</font></i>: 
( <font color="Maroon" title="c6">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">R</font> &amp; <font color="Maroon" title="c5">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">L</font> )
 ;<br/><a NAME="E2:28_3_5"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_5_1"/><i><font color="Green" title="E40">A76</font></i>: 
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_5_1_1"/><i><font color="Green" title="E41">A77</font></i>: 
<font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_5_1_1_1"/><i><font color="Green" title="E42">A78</font></i>: 
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c19">e</font> = <font color="Maroon" title="c18">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<a NAME="E3:28_3_5_1_1_1"/><i><font color="Green" title="E43">A79</font></i>: 
 ex <font color="Olive" title="b1">e'</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c19">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e'</font> &amp; <font color="Olive" title="b1">e'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_1_1"><i><font color="Green" title="E42">A78</font></i></a></i>;<br/>
<a NAME="E4:28_3_5_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></i>;<br/>
<b>hence </b><a NAME="E5:28_3_5_1_1_1"/>
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E3:28_3_5_1_1_1"><i><font color="Green" title="E43">A79</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_5_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_5_1_1_2"/>
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:28_3_5_1_1_2"/><b>then </b><i><font color="Green" title="E42">A80</font></i>: 
<font color="Maroon" title="c18">e</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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">e</font> = <font color="Maroon" title="c18">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> ;<br/>
<a NAME="E4:28_3_5_1_1_2"/>
<font color="Maroon" title="c19">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_5_1_1_2"><i><font color="Green" title="E42">A80</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:28_3_5_1_1_2"/>
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_3_5_1_1"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_5_1_1"><i><font color="Green" title="E41">A77</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_5_1"/><i><font color="Green" title="E41">A81</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>;<br/>

<b>assume </b><a NAME="E1:28_3_5_1_2"/><i><font color="Green" title="E42">A82</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ;<br/>

<div><i><font color="Green" title="E43">A83</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3_5_1_2_1"/><i><font color="Green" title="E44">A84</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 ;<br/><a NAME="E2:28_3_5_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_5_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">c</font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28_3_5_1"><i><font color="Green" title="E40">A76</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2"><i><font color="Green" title="E42">A82</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_1"><i><font color="Green" title="E44">A84</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E2:28_3_5_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T27" target="_self" title="PROJRED1:th.27">Th27</a></i>;<br/>
<a NAME="E3:28_3_5_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T27" target="_self" title="PROJRED1:th.27">Th27</a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_5_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E22:28_3"><i><font color="Green" title="E36">A40</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_1"><i><font color="Green" title="E44">A84</font></i></a>, <a class="ref" href="projred1.html#T27" target="_self" title="PROJRED1:th.27">Th27</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_3_5_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E44">A85</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3_5_1_2_2"/><i><font color="Green" title="E45">A86</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c1</font>
 ;<br/><a NAME="E2:28_3_5_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_5_1_2_2_1"/><i><font color="Green" title="E46">A87</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">c3</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28_3_5_1"><i><font color="Green" title="E40">A76</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2"><i><font color="Green" title="E42">A82</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_2"><i><font color="Green" title="E45">A86</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E2:28_3_5_1_2_2_1"/><i><font color="Green" title="E47">A88</font></i>: 
 ex <font color="Olive" title="b1">A1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A1</font> &amp; <font color="Maroon" title="c10">c1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A1</font> &amp; <font color="Maroon" title="c16">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A1</font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E5:28_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="projred1.html#E17:28_3"><i><font color="Green" title="E32">A34</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">A2</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E4:28_3_5_1_2_2_1"/><i><font color="Green" title="E48">A89</font></i>: 
( <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">A2</font> &amp; <font color="Maroon" title="c16">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">A2</font> &amp; <font color="Maroon" title="c17">c3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">A2</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E17:28_3"><i><font color="Green" title="E32">A34</font></i></a>, <a class="txt" href="projred1.html#E19:28_3"><i><font color="Green" title="E33">A35</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<a NAME="E5:28_3_5_1_2_2_1"/>
<font color="Maroon" title="c16">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E3:28_3"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="projred1.html#E5:28_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_5_1_2_2_1"><i><font color="Green" title="E47">A88</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E6:28_3_5_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">A2</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E3:28_3"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="projred1.html#E17:28_3"><i><font color="Green" title="E32">A34</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_5_1_2_2_1"><i><font color="Green" title="E48">A89</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E7:28_3_5_1_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <b>by </b><i><a class="txt" href="projred1.html#E5:28_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="projred1.html#E15:28_3"><i><font color="Green" title="E31">A33</font></i></a>, <a class="txt" href="projred1.html#E19:28_3"><i><font color="Green" title="E33">A35</font></i></a>, <a class="txt" href="projred1.html#E22:28_3"><i><font color="Green" title="E36">A40</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_2"><i><font color="Green" title="E45">A86</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_2_1"><i><font color="Green" title="E46">A87</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_5_1_2_2_1"><i><font color="Green" title="E48">A89</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_3_5_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E45">A90</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3_5_1_2_3"/><i><font color="Green" title="E46">A91</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a1</font>
 ;<br/><a NAME="E2:28_3_5_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_5_1_2_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a3</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28_3_5_1"><i><font color="Green" title="E40">A76</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2"><i><font color="Green" title="E42">A82</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_3"><i><font color="Green" title="E46">A91</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E2:28_3_5_1_2_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <b>by </b><i><a class="txt" href="projred1.html#E7:28_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="projred1.html#E11:28_3"><i><font color="Green" title="E29">A31</font></i></a>, <a class="txt" href="projred1.html#E13:28_3"><i><font color="Green" title="E30">A32</font></i></a>, <a class="txt" href="projred1.html#E15:28_3"><i><font color="Green" title="E31">A33</font></i></a>, <a class="txt" href="projred1.html#E22:28_3"><i><font color="Green" title="E36">A40</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_3"><i><font color="Green" title="E46">A91</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_3_5_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3_5_1_2_4"/><i><font color="Green" title="E46">A92</font></i>: 
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c1</font> &amp; <font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">a1</font> )
 ;<br/><a NAME="E2:28_3_5_1_2_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_5_1_2_4_1"/><i><font color="Green" title="E47">A93</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></i>;<br/>
<a NAME="E2:28_3_5_1_2_4_1"/><b>then </b><i><font color="Green" title="E48">A94</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1"><i><font color="Green" title="E40">A76</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2"><i><font color="Green" title="E42">A82</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">y</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T22" target="_self" title="PROJRED1:th.22">Th22</a></i>;<br/>
<a NAME="E4:28_3_5_1_2_4_1"/><i><font color="Green" title="E49">A95</font></i>: 
<font color="Maroon" title="c19">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_5_1_2_4_1"><i><font color="Green" title="E48">A94</font></i></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">z</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T22" target="_self" title="PROJRED1:th.22">Th22</a></i>;<br/>
<a NAME="E6:28_3_5_1_2_4_1"/><i><font color="Green" title="E50">A96</font></i>: 
<font color="Maroon" title="c20">z</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_5_1_2_4_1"><i><font color="Green" title="E49">A95</font></i></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">A1</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E8:28_3_5_1_2_4_1"/><i><font color="Green" title="E51">A97</font></i>: 
( <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">A1</font> &amp; <font color="Maroon" title="c12">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">A1</font> &amp; <font color="Maroon" title="c13">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">A1</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E9:28_3"><i><font color="Green" title="E28">A30</font></i></a>, <a class="txt" href="projred1.html#E11:28_3"><i><font color="Green" title="E29">A31</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">A3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E10:28_3_5_1_2_4_1"/><i><font color="Green" title="E52">A98</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c22">A3</font> &amp; <font color="Maroon" title="c11">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c22">A3</font> &amp; <font color="Maroon" title="c12">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c22">A3</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="projred1.html#E9:28_3"><i><font color="Green" title="E28">A30</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c23">B1</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E12:28_3_5_1_2_4_1"/><i><font color="Green" title="E53">A99</font></i>: 
( <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c23">B1</font> &amp; <font color="Maroon" title="c19">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c23">B1</font> &amp; <font color="Maroon" title="c20">z</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c23">B1</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_5_1_2_4_1"><i><font color="Green" title="E49">A95</font></i></a>, <a class="txt" href="projred1.html#E6:28_3_5_1_2_4_1"><i><font color="Green" title="E50">A96</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">B3</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E14:28_3_5_1_2_4_1"/><i><font color="Green" title="E54">A100</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c24">B3</font> &amp; <font color="Maroon" title="c18">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c24">B3</font> &amp; <font color="Maroon" title="c19">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c24">B3</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_5_1_2_4_1"><i><font color="Green" title="E48">A94</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_5_1_2_4_1"><i><font color="Green" title="E49">A95</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<a NAME="E15:28_3_5_1_2_4_1"/><i><font color="Green" title="E55">A101</font></i>: 
( <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; <font color="Maroon" title="c18">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; <font color="Maroon" title="c11">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1"><i><font color="Green" title="E40">A76</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2"><i><font color="Green" title="E42">A82</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_4_1"><i><font color="Green" title="E47">A93</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<a NAME="E16:28_3_5_1_2_4_1"/><i><font color="Green" title="E56">A102</font></i>: 
( <font color="Maroon" title="c15">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">B</font> &amp; <font color="Maroon" title="c13">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">B</font> &amp; <font color="Maroon" title="c11">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">B</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; <font color="Maroon" title="c18">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; <font color="Maroon" title="c11">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <font color="Maroon" title="c20">z</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <font color="Maroon" title="c13">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="projred1.html#E9:28_3"><i><font color="Green" title="E28">A30</font></i></a>, <a class="txt" href="projred1.html#E13:28_3"><i><font color="Green" title="E30">A32</font></i></a>, <a class="txt" href="projred1.html#E15:28_3"><i><font color="Green" title="E31">A33</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1"><i><font color="Green" title="E40">A76</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2"><i><font color="Green" title="E42">A82</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_4_1"><i><font color="Green" title="E47">A93</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_5_1_2_4_1"><i><font color="Green" title="E49">A95</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></i>;<br/>
<a NAME="E17:28_3_5_1_2_4_1"/><i><font color="Green" title="E57">A103</font></i>: 
<font color="Maroon" title="c21">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">A3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_4_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_3_5_1_2_4_1_1"/>
<font color="Maroon" title="c21">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">A3</font>
 ;<br/>

<a NAME="E2:28_3_5_1_2_4_1_1"/><b>then </b>
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">A3</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28_3"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="projred1.html#E3:28_3"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="projred1.html#E8:28_3_5_1_2_4_1"><i><font color="Green" title="E51">A97</font></i></a>, <a class="txt" href="projred1.html#E10:28_3_5_1_2_4_1"><i><font color="Green" title="E52">A98</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:28_3_5_1_2_4_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E5:28_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="projred1.html#E10:28_3_5_1_2_4_1"><i><font color="Green" title="E52">A98</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:28_3_5_1_2_4_1"/><i><font color="Green" title="E58">A104</font></i>: 
<font color="Maroon" title="c12">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_4_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_3_5_1_2_4_1_2"/><i><font color="Green" title="E59">A105</font></i>: 
<font color="Maroon" title="c12">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 ;<br/>

<a NAME="E2:28_3_5_1_2_4_1_2"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T27" target="_self" title="PROJRED1:th.27">Th27</a></i>;<br/>
<b>hence </b><a NAME="E3:28_3_5_1_2_4_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_4_1_2"><i><font color="Green" title="E59">A105</font></i></a>, <a class="ref" href="projred1.html#T26" target="_self" title="PROJRED1:th.26">Th26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:28_3_5_1_2_4_1"/><i><font color="Green" title="E59">A106</font></i>: 
<font color="Maroon" title="c12">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">a3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_4_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_3_5_1_2_4_1_3"/>
<font color="Maroon" title="c12">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a3</font>
 ;<br/>

<a NAME="E2:28_3_5_1_2_4_1_3"/><b>then </b><i><font color="Green" title="E60">A107</font></i>: 
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E9:28_3"><i><font color="Green" title="E28">A30</font></i></a>, <a class="txt" href="projred1.html#E11:28_3"><i><font color="Green" title="E29">A31</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5"><i><font color="Green" title="E39">A75</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E3:28_3_5_1_2_4_1_3"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="projred1.html#T27" target="_self" title="PROJRED1:th.27">Th27</a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_5_1_2_4_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_5_1_2_4_1_3"><i><font color="Green" title="E60">A107</font></i></a>, <a class="ref" href="projred1.html#T26" target="_self" title="PROJRED1:th.26">Th26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:28_3_5_1_2_4_1"/><i><font color="Green" title="E60">A108</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">y</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_4"><i><font color="Green" title="E46">A92</font></i></a>, <a class="txt" href="projred1.html#E14:28_3_5_1_2_4_1"><i><font color="Green" title="E54">A100</font></i></a>, <a class="txt" href="projred1.html#E15:28_3_5_1_2_4_1"><i><font color="Green" title="E55">A101</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E21:28_3_5_1_2_4_1"/>
( <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c12">a2</font>,<font color="Maroon" title="c19">y</font>,<font color="Maroon" title="c4">c</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c6">L</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c12">a2</font>,<font color="Maroon" title="c13">a3</font>,<font color="Maroon" title="c3">q</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c21">A1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c12">a2</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c11">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c22">A3</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c3">q</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c9">A</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c19">y</font>,<font color="Maroon" title="c18">x</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c24">B3</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c19">y</font>,<font color="Maroon" title="c20">z</font>,<font color="Maroon" title="c3">q</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c23">B1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c13">a3</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c11">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c14">B</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c11">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c5">K</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c13">a3</font>,<font color="Maroon" title="c20">z</font>,<font color="Maroon" title="c4">c</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c7">R</font> &amp; <font color="Maroon" title="c21">A1</font>,<font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c22">A3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a>  &amp; <font color="Maroon" title="c12">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c12">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c12">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">a3</font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred1.html#E3:28_3"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="projred1.html#E9:28_3"><i><font color="Green" title="E28">A30</font></i></a>, <a class="txt" href="projred1.html#E15:28_3"><i><font color="Green" title="E31">A33</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_5_1_2_4_1"><i><font color="Green" title="E49">A95</font></i></a>, <a class="txt" href="projred1.html#E8:28_3_5_1_2_4_1"><i><font color="Green" title="E51">A97</font></i></a>, <a class="txt" href="projred1.html#E10:28_3_5_1_2_4_1"><i><font color="Green" title="E52">A98</font></i></a>, <a class="txt" href="projred1.html#E12:28_3_5_1_2_4_1"><i><font color="Green" title="E53">A99</font></i></a>, <a class="txt" href="projred1.html#E14:28_3_5_1_2_4_1"><i><font color="Green" title="E54">A100</font></i></a>, <a class="txt" href="projred1.html#E16:28_3_5_1_2_4_1"><i><font color="Green" title="E56">A102</font></i></a>, <a class="txt" href="projred1.html#E17:28_3_5_1_2_4_1"><i><font color="Green" title="E57">A103</font></i></a>, <a class="txt" href="projred1.html#E18:28_3_5_1_2_4_1"><i><font color="Green" title="E58">A104</font></i></a>, <a class="txt" href="projred1.html#E19:28_3_5_1_2_4_1"><i><font color="Green" title="E59">A106</font></i></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c25">O</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E23:28_3_5_1_2_4_1"/><i><font color="Green" title="E61">A109</font></i>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c20">z</font>,<font color="Maroon" title="c18">x</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c25">O</font>
 <b>by </b><i><a class="txt" href="projred1.html#E20:28_3_5_1_2_4_1"><i><font color="Green" title="E60">A108</font></i></a>, <a class="ref" href="projred1.html#T14" target="_self" title="PROJRED1:th.14">Th14</a></i>;<br/>
<a NAME="E24:28_3_5_1_2_4_1"/><i><font color="Green" title="E62">A110</font></i>: 
( <font color="Maroon" title="c18">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">O</font> &amp; <font color="Maroon" title="c20">z</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">O</font> &amp; <font color="Maroon" title="c15">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">O</font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E23:28_3_5_1_2_4_1"><i><font color="Green" title="E61">A109</font></i></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></i>;<br/>
<a NAME="E25:28_3_5_1_2_4_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28_3_5_1"><i><font color="Green" title="E40">A76</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2"><i><font color="Green" title="E42">A82</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E26:28_3_5_1_2_4_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <b>by </b><i><a class="txt" href="projred1.html#E22:28_3"><i><font color="Green" title="E36">A40</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_5_1_2_4_1"><i><font color="Green" title="E48">A94</font></i></a>, <a class="txt" href="projred1.html#E6:28_3_5_1_2_4_1"><i><font color="Green" title="E50">A96</font></i></a>, <a class="txt" href="projred1.html#E24:28_3_5_1_2_4_1"><i><font color="Green" title="E62">A110</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_3_5_1_2_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:28_3_5_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <b>by </b><i><a class="txt" href="projred1.html#E2:28_3_5_1_2"><i><font color="Green" title="E43">A83</font></i></a>, <a class="txt" href="projred1.html#E3:28_3_5_1_2"><i><font color="Green" title="E44">A85</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_5_1_2"><i><font color="Green" title="E45">A90</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:28_3_5_1"/><i><font color="Green" title="E42">A111</font></i>: 
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_5_1_3"/><i><font color="Green" title="E43">A112</font></i>: 
<font color="Maroon" title="c8">X</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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_5_1_3_1"/><i><font color="Green" title="E44">A113</font></i>: 
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c19">e</font> = <font color="Maroon" title="c18">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<a NAME="E3:28_3_5_1_3_1"/>
 ex <font color="Olive" title="b1">e'</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c19">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e'</font> &amp; <font color="Olive" title="b1">e'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_3_1"><i><font color="Green" title="E44">A113</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_5_1_3_1"/>
<font color="Maroon" title="c18">e</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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E22:28_3"><i><font color="Green" title="E36">A40</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_5_1_3"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_5_1_3_2"/><i><font color="Green" title="E44">A114</font></i>: 
<font color="Maroon" title="c18">e</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> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c19">e</font> = <font color="Maroon" title="c18">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> ;<br/>
<a NAME="E3:28_3_5_1_3_2"/>
<font color="Maroon" title="c19">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E22:28_3"><i><font color="Green" title="E36">A40</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_3_2"><i><font color="Green" title="E44">A114</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_5_1_3_2"/>
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_3_5_1_3"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_5_1_3"><i><font color="Green" title="E43">A112</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:28_3_5_1"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>,the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:28"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>hence </b><a NAME="E5:28_3_5_1"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">R</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_5_1"><i><font color="Green" title="E40">A76</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_5_1"><i><font color="Green" title="E41">A81</font></i></a>, <a class="txt" href="projred1.html#E3:28_3_5_1"><i><font color="Green" title="E42">A111</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_3_5"/>
 ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E22:28_3"><i><font color="Green" title="E36">A40</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E26:28_3"/>
 ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E23:28_3"><i><font color="Green" title="E37">A48</font></i></a>, <a class="txt" href="projred1.html#E24:28_3"><i><font color="Green" title="E38">A61</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:28"/>
 ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E5:28"><i><font color="Green" title="E23">A6</font></i></a></i>;<br/>


</div>
