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

<b>let </b><font color="Maroon">IPP</font> be  <a href="incproj.html#V6">2-dimensional</a> <a href="incproj.html#V10">Desarguesian</a> <a href="incproj.html#NM2">IncProjSp</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">K</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">L</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/>













<b>assume </b><a NAME="E1:28"/><i><font color="Green">E33</font></i>: 
( not <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> &amp; not <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">L</font> &amp; not <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">L</font> &amp; not <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">L</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> &amp; <font color="Maroon">K</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">R</font> )
 ;<br/>

<a NAME="E2:28"/><i><font color="Green">E34</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">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">f</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span>;<br/>
<a NAME="E1:28_1"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:28_1"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E3:28_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E4:28_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font>
 <b>by </b><i>, <a class="ref" href="relset_1.html#T12">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">set</a> ] <b>means </b> ex <font color="Olive">k</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> );<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:28"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<div><i><font color="Green">E39</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">E40</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/><a NAME="E2:28_2"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">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">E41</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">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">X</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">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">E62</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">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">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_2_1_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> <b>by </b><i/>;<br/>
<a NAME="E3:28_2_1_1_1_1"/><i><font color="Green">E64</font></i>: 
 ex <font color="Olive">e'</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">e'</font> &amp; <font color="Olive">e'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:28_2_1_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:28_2_1_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_2_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">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">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_2_1_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:28_2_1_1_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> ;<br/>
<a NAME="E4:28_2_1_1_1_2"/>
<font color="Maroon">e</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E5:28_2_1_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_2_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">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">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">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">E66</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">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">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_2_1_1_2_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> <b>by </b><i/>;<br/>
<a NAME="E3:28_2_1_1_2_1"/>
 ex <font color="Olive">e'</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">e'</font> &amp; <font color="Olive">e'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:28_2_1_1_2_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_2_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">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">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_2_1_1_2_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> ;<br/>
<a NAME="E3:28_2_1_1_2_2"/>
<font color="Maroon">e</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E4:28_2_1_1_2_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_2_1_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">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">E69</font></i>: 
for <font color="Olive">x</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">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">x</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/>

<b>assume </b><a NAME="E1:28_2_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:28_2_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E3:28_2_1_2"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> <b>by </b><i>, , </i>;<br/>
<a NAME="E5:28_2_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">y</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">L</font>
 
<b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">A</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E7:28_2_1_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">B</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E9:28_2_1_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">y</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> )
 <b>by </b><i>, , , </i>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> <b>by </b><i>, , , </i>;<br/>
<a NAME="E11:28_2_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">z</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>then consider </b><font color="Maroon">C</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E13:28_2_1_2"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; <font color="Maroon">y</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; <font color="Maroon">z</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , </i>;<br/>
<a NAME="E14:28_2_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, , , <a class="ref" href="projred1.html#T1" target="_self">Th1</a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E15:28_2_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , , <a class="txt" href="projred1.html#E2:28"><i><font color="Green">E34</font></i></a>, <a class="ref" href="incproj.html#D9">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , , <a class="ref" href="projred1.html#T1" target="_self">Th1</a>, <a class="txt" href="projred1.html#E1:28"><i><font color="Green">E33</font></i></a>, <a class="txt" href="projred1.html#E2:28"><i><font color="Green">E34</font></i></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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> is   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font>,the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font>
 
<b>by </b><i>, <a class="ref" href="relset_1.html#T11">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font>
 <b>by </b><i>, , <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_2"/>
 ex <font color="Olive">o</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( not <font color="Olive">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> &amp; not <font color="Olive">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Olive">o</font>,<font color="Maroon">R</font> )
 <b>by </b><i>, </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">E79</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 ;<br/><b>consider </b><font color="Maroon">A</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E3:28_3"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/><b>consider </b><font color="Maroon">c1</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E5:28_3"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">c1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> &amp; <font color="Maroon">c1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14">INCPROJ:def 14</a></i>;<br/><b>consider </b><font color="Maroon">a1</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E7:28_3"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">a1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> &amp; <font color="Maroon">c1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">a2</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> <b>by </b><i>, , </i>;<br/><a NAME="E9:28_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">L</font>
 <b>by </b><i>, , </i>;<br/><b>then reconsider </b><font color="Maroon">a3</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> <b>by </b><i>, </i>;<br/><a NAME="E11:28_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">a3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font>
 <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">B</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E13:28_3"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">a1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/><b>consider </b><font color="Maroon">o</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E15:28_3"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14">INCPROJ:def 14</a></i>;<br/><b>reconsider </b><font color="Maroon">c2</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c1</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font> <b>by </b><i>, , </i>;<br/><a NAME="E17:28_3"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">L</font>
 <b>by </b><i>, , </i>;<br/><b>then reconsider </b><font color="Maroon">c3</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c2</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> <b>by </b><i>, </i>;<br/><a NAME="E19:28_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="projred1.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E20:28_3"/><i><font color="Green">E89</font></i>: 
not <font color="Maroon">a1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font>
 <b>by </b><i>, , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E21:28_3"/><i><font color="Green">E90</font></i>: 
not <font color="Maroon">a3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">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">a3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font>
 ;<br/>

<a NAME="E2:28_3_1"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font>
 
<b>by </b><i>, , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E3:28_3_1"/>
 ex <font color="Olive">C</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">C</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">C</font> )
 
<b>by </b><i><a class="ref" href="incproj.html#D10">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font>
 
<b>by </b><i>, <a class="ref" href="projred1.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E5:28_3_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:28_3_1"/>
 ex <font color="Olive">D</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">D</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">D</font> )
 
<b>by </b><i><a class="ref" href="incproj.html#D10">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" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E8:28_3_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E22:28_3"/><i><font color="Green">E93</font></i>: 
( not <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> &amp; not <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">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">E94</font></i>: 
( <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> or <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> )
 ;<br/>

<div><i><font color="Green">E95</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">E96</font></i>: 
<font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E2:28_3_2_1"/><b>then </b>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">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#E5:28"><i><font color="Green">E39</font></i></a>, <a class="ref" href="incproj.html#D9">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">E97</font></i>: 
<font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font>
 ;<br/><a NAME="E2:28_3_2_2"/><i><font color="Green">E98</font></i>: 
 ex <font color="Olive">A1</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">A1</font> &amp; <font color="Maroon">c1</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">A1</font> &amp; <font color="Maroon">c2</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">A1</font> )
 <b>by </b><i>, , <a class="ref" href="projred1.html#T1" target="_self">Th1</a>, </i>;<br/><b>consider </b><font color="Maroon">A2</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E4:28_3_2_2"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">c2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">c3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A2</font> )
 <b>by </b><i>, <a class="ref" href="projred1.html#T1" target="_self">Th1</a>, <a class="txt" href="projred1.html#E1:28"><i><font color="Green">E33</font></i></a>, </i>;<br/><a NAME="E5:28_3_2_2"/>
<font color="Maroon">c2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E6:28_3_2_2"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 <b>by </b><i>, , <a class="ref" href="projred1.html#T1" target="_self">Th1</a>, , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E7:28_3_2_2"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c3</font>
 <b>by </b><i>, , <a class="txt" href="projred1.html#E1:28"><i><font color="Green">E33</font></i></a>, <a class="txt" href="projred1.html#E5:28"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E8:28_3_2_2"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font>
 <b>by </b><i>, , , <a class="txt" href="projred1.html#E2:28"><i><font color="Green">E34</font></i></a>, <a class="txt" href="projred1.html#E5:28"><i><font color="Green">E39</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E9:28_3_2_2"/><b>then </b>
<font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 <b>by </b><i>, , <a class="ref" href="projred1.html#T1" target="_self">Th1</a>, <a class="ref" href="projred1.html#T3" target="_self">Th3</a>, </i>;<br/><b>hence </b><a NAME="E10:28_3_2_2"/>
contradiction
 <b>by </b><i>, , , </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#E4:28"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E101</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">E102</font></i>: 
<font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font>
 ;<br/><b>take </b><font color="Maroon">q</font> = <font color="Maroon">q</font>;<br/><a NAME="E2:28_3_3"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">q</font>,<font color="Maroon">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">E103</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">q</font>,<font color="Maroon">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">X</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">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">E104</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">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">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_3_1_1_1_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font> <b>by </b><i/>;<br/>
<a NAME="E3:28_3_3_1_1_1_1"/><i><font color="Green">E106</font></i>: 
 ex <font color="Olive">e'</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">e'</font> &amp; <font color="Olive">e'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:28_3_3_1_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:28_3_3_1_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="projred1.html#T3" target="_self">Th3</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_3_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">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">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_3_1_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:28_3_3_1_1_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> ;<br/>
<a NAME="E4:28_3_3_1_1_1_2"/>
<font color="Maroon">e</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E5:28_3_3_1_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_3_3_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">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">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">q</font>,<font color="Maroon">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">E108</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">q</font>,<font color="Maroon">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">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_3_1_1_2_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> <b>by </b><i/>;<br/>
<a NAME="E3:28_3_3_1_1_2_1"/>
 ex <font color="Olive">e'</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">e'</font> &amp; <font color="Olive">e'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:28_3_3_1_1_2_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="projred1.html#E4:28"><i><font color="Green">E38</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_3_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">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">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_3_1_1_2_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> ;<br/>
<a NAME="E3:28_3_3_1_1_2_2"/>
<font color="Maroon">e</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="txt" href="projred1.html#E4:28"><i><font color="Green">E38</font></i></a>, , </i>;<br/>
<b>hence </b><a NAME="E4:28_3_3_1_1_2_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_3_3_1_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">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">E111</font></i>: 
for <font color="Olive">x</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">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">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font>;<br/>

<b>assume </b><a NAME="E1:28_3_3_1_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:28_3_3_1_2"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="txt" href="projred1.html#E4:28"><i><font color="Green">E38</font></i></a>, <a class="txt" href="projred1.html#E5:28"><i><font color="Green">E39</font></i></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" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">K</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E38</font></i></a>, <a class="txt" href="projred1.html#E5:28"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="funct_1.html#T22">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="projred1.html#E4:28"><i><font color="Green">E38</font></i></a>, <a class="ref" href="projred1.html#T3" target="_self">Th3</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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> is   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font>,the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font>
 
<b>by </b><i>, <a class="ref" href="relset_1.html#T11">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="projred1.html#E5:28"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_3_3"/>
 ex <font color="Olive">o</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( not <font color="Olive">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> &amp; not <font color="Olive">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Olive">o</font>,<font color="Maroon">R</font> )
 <b>by </b><i>, <a class="txt" href="projred1.html#E4:28"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E114</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">E115</font></i>: 
<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font>
 ;<br/><b>take </b><font color="Maroon">p</font> = <font color="Maroon">p</font>;<br/><a NAME="E2:28_3_4"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">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">E116</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">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">X</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">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">E117</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">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">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_4_1_1_1_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font> <b>by </b><i/>;<br/>
<a NAME="E3:28_3_4_1_1_1_1"/><i><font color="Green">E119</font></i>: 
 ex <font color="Olive">e'</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">e'</font> &amp; <font color="Olive">e'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> )
 
<b>by </b><i>, <a class="ref" href="projred1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:28_3_4_1_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:28_3_4_1_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_4_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">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">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_4_1_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:28_3_4_1_1_1_2"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> ;<br/>
<a NAME="E4:28_3_4_1_1_1_2"/>
<font color="Maroon">e</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="ref" href="projred1.html#T3" target="_self">Th3</a>, </i>;<br/>
<b>hence </b><a NAME="E5:28_3_4_1_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_3_4_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">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">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">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">E121</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">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">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_4_1_1_2_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> <b>by </b><i/>;<br/>
<a NAME="E3:28_3_4_1_1_2_1"/>
 ex <font color="Olive">e'</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">e'</font> &amp; <font color="Olive">e'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> )
 
<b>by </b><i>, <a class="ref" href="projred1.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_4_1_1_2_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="projred1.html#E5:28"><i><font color="Green">E39</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_4_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">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">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_4_1_1_2_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> ;<br/>
<a NAME="E3:28_3_4_1_1_2_2"/>
<font color="Maroon">e</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="txt" href="projred1.html#E5:28"><i><font color="Green">E39</font></i></a>, <a class="ref" href="projred1.html#T3" target="_self">Th3</a>, </i>;<br/>
<b>hence </b><a NAME="E4:28_3_4_1_1_2_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_3_4_1_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">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">E124</font></i>: 
for <font color="Olive">x</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">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">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font>;<br/>

<b>assume </b><a NAME="E1:28_3_4_1_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:28_3_4_1_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="txt" href="projred1.html#E5:28"><i><font color="Green">E39</font></i></a>, , </i>;<br/>
<a NAME="E3:28_3_4_1_2"/><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">R</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">R</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E5:28"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="projred1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> <b>by </b><i>, <a class="txt" href="projred1.html#E5:28"><i><font color="Green">E39</font></i></a>, , </i>;<br/>
<a NAME="E5:28_3_4_1_2"/>
<font color="Maroon">y</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="txt" href="projred1.html#E5:28"><i><font color="Green">E39</font></i></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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="projred1.html#E5:28"><i><font color="Green">E39</font></i></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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> is   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font>,the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font>
 
<b>by </b><i>, <a class="ref" href="relset_1.html#T11">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">R</font>
 <b>by </b><i>, , <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_3_4"/>
 ex <font color="Olive">o</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( not <font color="Olive">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> &amp; not <font color="Olive">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Olive">o</font>,<font color="Maroon">R</font> )
 <b>by </b><i>, <a class="txt" href="projred1.html#E5:28"><i><font color="Green">E39</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">E128</font></i>: 
( <font color="Maroon">L</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">R</font> &amp; <font color="Maroon">K</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">L</font> )
 ;<br/><a NAME="E2:28_3_5"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">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">E129</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">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">E130</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">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">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_5_1_1_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> <b>by </b><i/>;<br/>
<a NAME="E3:28_3_5_1_1_1"/><i><font color="Green">E132</font></i>: 
 ex <font color="Olive">e'</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">e'</font> &amp; <font color="Olive">e'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> )
 
<b>by </b><i>, <a class="ref" href="projred1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:28_3_5_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:28_3_5_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_5_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">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">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_5_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:28_3_5_1_1_2"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> ;<br/>
<a NAME="E4:28_3_5_1_1_2"/>
<font color="Maroon">e</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="ref" href="projred1.html#T3" target="_self">Th3</a>, </i>;<br/>
<b>hence </b><a NAME="E5:28_3_5_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_3_5_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_5_1"/><i><font color="Green">E134</font></i>: 
for <font color="Olive">x</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">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">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font>;<br/>

<b>assume </b><a NAME="E1:28_3_5_1_2"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<div><i><font color="Green">E136</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">E137</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="projred1.html#T3" target="_self">Th3</a>, , <a class="ref" href="funct_1.html#T22">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
 
<b>by </b><i>, </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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
<b>by </b><i>, </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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="projred1.html#T2" target="_self">Th2</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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E138</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">E139</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">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">E140</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c3</font>
 
<b>by </b><i>, <a class="ref" href="projred1.html#T3" target="_self">Th3</a>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E2:28_3_5_1_2_2_1"/><i><font color="Green">E141</font></i>: 
 ex <font color="Olive">A1</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">A1</font> &amp; <font color="Maroon">c1</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">A1</font> &amp; <font color="Maroon">c2</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">A1</font> )
 
<b>by </b><i>, , <a class="ref" href="projred1.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>consider </b><font color="Maroon">A2</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E4:28_3_5_1_2_2_1"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">c2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">c3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A2</font> )
 <b>by </b><i>, <a class="ref" href="projred1.html#T1" target="_self">Th1</a>, <a class="txt" href="projred1.html#E1:28"><i><font color="Green">E33</font></i></a>, </i>;<br/>
<a NAME="E5:28_3_5_1_2_2_1"/>
<font color="Maroon">c2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E6:28_3_5_1_2_2_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 
<b>by </b><i>, , <a class="ref" href="projred1.html#T1" target="_self">Th1</a>, , <a class="ref" href="incproj.html#D9">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="txt" href="projred1.html#E1:28"><i><font color="Green">E33</font></i></a>, <a class="ref" href="projred1.html#T2" target="_self">Th2</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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E143</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">E144</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font>
 
<b>by </b><i>, <a class="ref" href="projred1.html#T3" target="_self">Th3</a>, , <a class="ref" href="funct_1.html#T22">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , , <a class="ref" href="projred1.html#T2" target="_self">Th2</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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">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">E145</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">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">E146</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:28_3_5_1_2_4_1"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font>
 
<b>by </b><i>, , <a class="ref" href="projred1.html#T3" target="_self">Th3</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> <b>by </b><i>, </i>;<br/>
<a NAME="E4:28_3_5_1_2_4_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">y</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">L</font>
 
<b>by </b><i>, , </i>;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> <b>by </b><i>, </i>;<br/>
<a NAME="E6:28_3_5_1_2_4_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">z</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">A1</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E8:28_3_5_1_2_4_1"/><i><font color="Green">E150</font></i>: 
( <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">a3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A1</font> )
 <b>by </b><i>, , , </i>;<br/>
<b>consider </b><font color="Maroon">A3</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E10:28_3_5_1_2_4_1"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A3</font> &amp; <font color="Maroon">a1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A3</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A3</font> )
 <b>by </b><i>, , , </i>;<br/>
<b>consider </b><font color="Maroon">B1</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E12:28_3_5_1_2_4_1"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">y</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">z</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B1</font> )
 <b>by </b><i>, , <a class="ref" href="projred1.html#T7" target="_self">Th7</a>, </i>;<br/>
<b>consider </b><font color="Maroon">B3</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E14:28_3_5_1_2_4_1"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B3</font> &amp; <font color="Maroon">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B3</font> &amp; <font color="Maroon">y</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B3</font> )
 <b>by </b><i>, , , </i>;<br/>
<a NAME="E15:28_3_5_1_2_4_1"/><i><font color="Green">E193</font></i>: 
( <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> &amp; <font color="Maroon">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> &amp; <font color="Maroon">a1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> )
 
<b>by </b><i>, , , <a class="ref" href="projred1.html#T3" target="_self">Th3</a>, , </i>;<br/>
<a NAME="E16:28_3_5_1_2_4_1"/><i><font color="Green">E194</font></i>: 
( <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> &amp; <font color="Maroon">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> &amp; <font color="Maroon">a1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> &amp; <font color="Maroon">z</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> &amp; <font color="Maroon">a3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> )
 
<b>by </b><i>, , , , , , <a class="ref" href="projred1.html#T3" target="_self">Th3</a>, , , , </i>;<br/>
<a NAME="E17:28_3_5_1_2_4_1"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">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">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A3</font>
 ;<br/>

<a NAME="E2:28_3_5_1_2_4_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A3</font>
 
<b>by </b><i>, , <a class="ref" href="projred1.html#T8" target="_self">Th8</a>, <a class="ref" href="projred1.html#T12" target="_self">Th12</a>, <a class="ref" href="incproj.html#D9">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="ref" href="projred1.html#T12" target="_self">Th12</a>, <a class="ref" href="incproj.html#D9">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">E196</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">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">E197</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:28_3_5_1_2_4_1_2"/>
contradiction
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:28_3_5_1_2_4_1"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">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">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font>
 ;<br/>

<a NAME="E2:28_3_5_1_2_4_1_3"/><b>then </b><i><font color="Green">E199</font></i>: 
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
<b>by </b><i>, , , <a class="txt" href="projred1.html#E5:28"><i><font color="Green">E39</font></i></a>, <a class="ref" href="incproj.html#D9">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" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:28_3_5_1_2_4_1_3"/>
contradiction
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:28_3_5_1_2_4_1"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , <a class="ref" href="projred1.html#T14" target="_self">Th14</a>, , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E21:28_3_5_1_2_4_1"/>
( <font color="Maroon">a2</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font> <a href="incproj.html#R4">on</a> <font color="Maroon">L</font> &amp; <font color="Maroon">a2</font>,<font color="Maroon">a3</font>,<font color="Maroon">q</font> <a href="incproj.html#R4">on</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">a2</font>,<font color="Maroon">p</font>,<font color="Maroon">a1</font> <a href="incproj.html#R4">on</a> <font color="Maroon">A3</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">o</font>,<font color="Maroon">q</font> <a href="incproj.html#R4">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="incproj.html#R4">on</a> <font color="Maroon">B3</font> &amp; <font color="Maroon">y</font>,<font color="Maroon">z</font>,<font color="Maroon">q</font> <a href="incproj.html#R4">on</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">a3</font>,<font color="Maroon">o</font>,<font color="Maroon">a1</font> <a href="incproj.html#R4">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">c</font>,<font color="Maroon">a1</font> <a href="incproj.html#R4">on</a> <font color="Maroon">K</font> &amp; <font color="Maroon">a3</font>,<font color="Maroon">z</font>,<font color="Maroon">c</font> <a href="incproj.html#R4">on</a> <font color="Maroon">R</font> &amp; <font color="Maroon">A1</font>,<font color="Maroon">L</font>,<font color="Maroon">A3</font> <a href="incproj.html#R1">are_mutually_different</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3</font> )
 
<b>by </b><i>, , , , , <a class="ref" href="projred1.html#T8" target="_self">Th8</a>, <a class="ref" href="projred1.html#T12" target="_self">Th12</a>, <a class="ref" href="projred1.html#T13" target="_self">Th13</a>, <a class="ref" href="projred1.html#T14" target="_self">Th14</a>, , , , , <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon">O</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E23:28_3_5_1_2_4_1"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">z</font>,<font color="Maroon">x</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E24:28_3_5_1_2_4_1"/><i><font color="Green">E202</font></i>: 
( <font color="Maroon">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; <font color="Maroon">z</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> )
 
<b>by </b><i>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="projred1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#T22">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="projred1.html#T2" target="_self">Th2</a>, , <a class="ref" href="projred1.html#T7" target="_self">Th7</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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:28_3_5_1"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">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">E204</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">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">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_5_1_3_1"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> <b>by </b><i/>;<br/>
<a NAME="E3:28_3_5_1_3_1"/>
 ex <font color="Olive">e'</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">e'</font> &amp; <font color="Olive">e'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:28_3_5_1_3_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="projred1.html#T2" target="_self">Th2</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_5_1_3"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">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">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_5_1_3_2"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> ;<br/>
<a NAME="E3:28_3_5_1_3_2"/>
<font color="Maroon">e</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="projred1.html#T2" target="_self">Th2</a>, , </i>;<br/>
<b>hence </b><a NAME="E4:28_3_5_1_3_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_3_5_1_3"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> is   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font>,the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font>
 
<b>by </b><i>, <a class="ref" href="relset_1.html#T11">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" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">o</font>,<font color="Maroon">R</font>
 <b>by </b><i>, , <a class="ref" href="projred1.html#T3" target="_self">Th3</a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_3_5"/>
 ex <font color="Olive">o</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( not <font color="Olive">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> &amp; not <font color="Olive">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Olive">o</font>,<font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="projred1.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E26:28_3"/>
 ex <font color="Olive">o</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( not <font color="Olive">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> &amp; not <font color="Olive">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Olive">o</font>,<font color="Maroon">R</font> )
 <b>by </b><i>, <a class="txt" href="projred1.html#E4:28"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:28"/>
 ex <font color="Olive">o</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( not <font color="Olive">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> &amp; not <font color="Olive">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">L</font>,<font color="Maroon">q</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Maroon">p</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">K</font>,<font color="Olive">o</font>,<font color="Maroon">R</font> )
 <b>by </b><i/>;<br/>


</div>
