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<span class="kw">let </span><font color="Maroon" title="c1">IPP</font> be   <a href="incproj.html#V5" title="INCPROJ:attr.5">2-dimensional</a>   <a href="incproj.html#V9" title="INCPROJ:attr.9">Desarguesian</a>  <a href="incproj.html#NM2" title="INCPROJ:NM.2">IncProjSp</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">c</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font><br/> for <font color="Olive" title="b4">K</font>, <font color="Olive" title="b5">L</font>, <font color="Olive" title="b6">R</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font>  st not <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">K</font> &amp; not <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b5">L</font> &amp; not <font color="Olive" title="b2">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b5">L</font> &amp; not <font color="Olive" title="b2">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b6">R</font> &amp; <font color="Olive" title="b3">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">K</font> &amp; <font color="Olive" title="b3">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b5">L</font> &amp; <font color="Olive" title="b3">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b6">R</font> &amp; <font color="Olive" title="b4">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">R</font> holds <br/> ex <font color="Olive" title="b7">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b7">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">K</font> &amp; not <font color="Olive" title="b7">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b6">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Olive" title="b5">L</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b6">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Olive" title="b4">K</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b5">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Olive" title="b4">K</font>,<font color="Olive" title="b7">o</font>,<font color="Olive" title="b6">R</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">c</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">K</font>, <font color="Olive" title="b2">L</font>, <font color="Olive" title="b3">R</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font>  st not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> &amp; not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">L</font> &amp; not <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">L</font> &amp; not <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">R</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">L</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">R</font> &amp; <font color="Olive" title="b1">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">R</font> holds <br/> ex <font color="Olive" title="b4">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b4">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> &amp; not <font color="Olive" title="b4">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Olive" title="b2">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Olive" title="b3">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b2">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Olive" title="b1">K</font>,<font color="Olive" title="b4">o</font>,<font color="Olive" title="b3">R</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">K</font>, <font color="Maroon" title="c6">L</font>, <font color="Maroon" title="c7">R</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font> &amp; not <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font> &amp; not <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <font color="Maroon" title="c5">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">R</font> implies  ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> ) )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 <span class="kw">and </span><br/><a NAME="E2:31"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font>
 <span class="kw">and </span><br/><a NAME="E3:31"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
not <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font>
 <span class="kw">and </span><br/><a NAME="E4:31"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
not <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font>
 <span class="kw">and </span><br/><a NAME="E5:31"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 <span class="kw">and </span><br/><a NAME="E6:31"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font>
 <span class="kw">and </span><br/><a NAME="E7:31"/><span class="lab"><font color="Green" title="E26">A7</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font>
 <span class="kw">and </span><br/><a NAME="E8:31"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<font color="Maroon" title="c5">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">k</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> );<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:31"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></span>
<a NAME="E11:31"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c9">f</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>;<br/>
<a NAME="E1:31_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:31_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font></span><br/>

<a NAME="E3:31_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:31_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E30">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E31">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:31_2_1"/><span class="lab"><font color="Green" title="E32">A13</font></span>: 
<font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> or <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2_1_1"/><span class="lab"><font color="Green" title="E33">A14</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">e</font> = <font color="Maroon" title="c9">e</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>;<br/>
<a NAME="E3:31_2_1_1"/><span class="lab"><font color="Green" title="E34">A15</font></span>: 
 ex <font color="Olive" title="b1">e9</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e9</font> &amp; <font color="Olive" title="b1">e9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E1:31_2_1_1"><span class="lab"><font color="Green" title="E33">A14</font></span></a></span>;<br/>
<a NAME="E4:31_2_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:31_2_1_1"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31_2_1_1"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:31_2_1"/><span class="lab"><font color="Green" title="E33">A16</font></span>: 
<font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b2">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b2">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b2">o</font>,<font color="Maroon" title="c7">R</font> )</span><br/><a NAME="E3:31_2_1"/><span class="lab"><font color="Green" title="E34">A17</font></span>: 
<font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> or <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2_1_2"/><span class="lab"><font color="Green" title="E35">A18</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">e</font> = <font color="Maroon" title="c9">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>;<br/>
<a NAME="E3:31_2_1_2"/>
 ex <font color="Olive" title="b1">e9</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e9</font> &amp; <font color="Olive" title="b1">e9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E1:31_2_1_2"><span class="lab"><font color="Green" title="E35">A18</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:31_2_1_2"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E2:31_2_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:31_2_1"/><span class="lab"><font color="Green" title="E35">A19</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2_1_3"/><span class="lab"><font color="Green" title="E36">A20</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span><br/>

<a NAME="E2:31_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A21</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c7">R</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_2_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E1:31_2_1"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>;<br/>
<a NAME="E3:31_2_1_3"/><span class="lab"><font color="Green" title="E38">A22</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E2:31_2_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E3:31_2_1"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E1:31_2_1_3"><span class="lab"><font color="Green" title="E36">A20</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">y</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E2:31_2_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="ref" href="projred1.html#T22" target="_self" title="PROJRED1:th.22">Th22</a></span>;<br/>
<a NAME="E5:31_2_1_3"/>
<font color="Maroon" title="c10">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E2:31_2_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E3:31_2_1_3"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:31_2_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E2:31_2_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E2:31_2_1_3"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="ref" href="projred1.html#T21" target="_self" title="PROJRED1:th.21">Th21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:31_2_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> or <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2_1_4"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font></span><br/>

<a NAME="E2:31_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A23</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">e</font> = <font color="Maroon" title="c9">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> ;<br/>
<a NAME="E4:31_2_1_4"/>
<font color="Maroon" title="c10">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E2:31_2_1_4"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:31_2_1_4"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:31_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A24</font></span>: 
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_2_1"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><a NAME="E7:31_2_1"/><span class="lab"><font color="Green" title="E37">A25</font></span>: 
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>,the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:31"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c9">p</font> = <font color="Maroon" title="c2">p</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> )</span><br/><a NAME="E8:31_2_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> or <font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2_1_5"/><span class="lab"><font color="Green" title="E38">A26</font></span>: 
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">e</font> = <font color="Maroon" title="c10">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> ;<br/>
<a NAME="E3:31_2_1_5"/>
<font color="Maroon" title="c11">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E2:31_2_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E1:31_2_1_5"><span class="lab"><font color="Green" title="E38">A26</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:31_2_1_5"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E9:31_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_2_1"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E10:31_2_1"/>
 ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E2:31_2_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E6:31_2_1"><span class="lab"><font color="Green" title="E36">A24</font></span></a>, <a class="txt" href="#E4:31_2_1"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="txt" href="#E7:31_2_1"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c9">A</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><span class="kw"> such that </span><br/><a NAME="E3:31_2"/><span class="lab"><font color="Green" title="E32">A27</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font>
 <span class="kw">and </span><br/><a NAME="E4:31_2"/><span class="lab"><font color="Green" title="E33">A28</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">c1</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font><span class="kw"> such that </span><br/><a NAME="E6:31_2"/><span class="lab"><font color="Green" title="E34">A29</font></span>: 
<font color="Maroon" title="c10">c1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 <span class="kw">and </span><br/><a NAME="E7:31_2"/><span class="lab"><font color="Green" title="E35">A30</font></span>: 
<font color="Maroon" title="c10">c1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incproj.html#D14" title="INCPROJ:def.14">INCPROJ:def 14</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">c2</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">c1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E6:31_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="ref" href="projred1.html#T22" target="_self" title="PROJRED1:th.22">Th22</a></span>;<br/><a NAME="E9:31_2"/><span class="lab"><font color="Green" title="E36">A31</font></span>: 
<font color="Maroon" title="c11">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E6:31_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">c3</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">c2</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="projred1.html#T22" target="_self" title="PROJRED1:th.22">Th22</a></span>;<br/><a NAME="E11:31_2"/><span class="lab"><font color="Green" title="E37">A32</font></span>: 
<font color="Maroon" title="c12">c3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E9:31_2"><span class="lab"><font color="Green" title="E36">A31</font></span></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">a1</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font><span class="kw"> such that </span><br/><a NAME="E13:31_2"/><span class="lab"><font color="Green" title="E38">A33</font></span>: 
<font color="Maroon" title="c13">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 <span class="kw">and </span><br/><a NAME="E14:31_2"/><span class="lab"><font color="Green" title="E39">A34</font></span>: 
<font color="Maroon" title="c10">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">a1</font>
 <span class="kw">and </span><br/><a NAME="E15:31_2"/><span class="lab"><font color="Green" title="E40">A35</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projred1.html#T8" target="_self" title="PROJRED1:th.8">Th8</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">a2</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a1</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E13:31_2"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="ref" href="projred1.html#T22" target="_self" title="PROJRED1:th.22">Th22</a></span>;<br/><a NAME="E17:31_2"/><span class="lab"><font color="Green" title="E41">A36</font></span>: 
<font color="Maroon" title="c14">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E13:31_2"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">a3</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a2</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="projred1.html#T22" target="_self" title="PROJRED1:th.22">Th22</a></span>;<br/><a NAME="E19:31_2"/><span class="lab"><font color="Green" title="E42">A37</font></span>: 
<font color="Maroon" title="c15">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E17:31_2"><span class="lab"><font color="Green" title="E41">A36</font></span></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></span>;<br/><a NAME="E20:31_2"/><span class="lab"><font color="Green" title="E43">A38</font></span>: 
not <font color="Maroon" title="c15">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:31_2_2"/>
<font color="Maroon" title="c15">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A39</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E7:31"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E8:31"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E19:31_2"><span class="lab"><font color="Green" title="E42">A37</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<a NAME="E3:31_2_2"/>
 ex <font color="Olive" title="b1">C</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></span>;<br/>
<a NAME="E4:31_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E7:31"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E2:31_2_2"><span class="lab"><font color="Green" title="E44">A39</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<a NAME="E5:31_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A40</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E17:31_2"><span class="lab"><font color="Green" title="E41">A36</font></span></a>, <a class="ref" href="projred1.html#T26" target="_self" title="PROJRED1:th.26">Th26</a></span>;<br/>
<a NAME="E6:31_2_2"/>
 ex <font color="Olive" title="b1">D</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">D</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></span>;<br/>
<a NAME="E7:31_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E5:31_2_2"><span class="lab"><font color="Green" title="E45">A40</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:31_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E13:31_2"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E15:31_2"><span class="lab"><font color="Green" title="E40">A35</font></span></a>, <a class="ref" href="projred1.html#T26" target="_self" title="PROJRED1:th.26">Th26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c16">B</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><span class="kw"> such that </span><br/><a NAME="E22:31_2"/><span class="lab"><font color="Green" title="E44">A41</font></span>: 
( <font color="Maroon" title="c13">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">B</font> &amp; <font color="Maroon" title="c15">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">o</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font><span class="kw"> such that </span><br/><a NAME="E24:31_2"/><span class="lab"><font color="Green" title="E45">A42</font></span>: 
<font color="Maroon" title="c17">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font>
 <span class="kw">and </span><br/><a NAME="E25:31_2"/><span class="lab"><font color="Green" title="E46">A43</font></span>: 
<font color="Maroon" title="c17">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incproj.html#D14" title="INCPROJ:def.14">INCPROJ:def 14</a></span>;<br/><a NAME="E26:31_2"/><span class="lab"><font color="Green" title="E47">A44</font></span>: 
not <font color="Maroon" title="c13">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E7:31"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E8:31"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E13:31_2"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E15:31_2"><span class="lab"><font color="Green" title="E40">A35</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/><a NAME="E27:31_2"/><span class="lab"><font color="Green" title="E48">A45</font></span>: 
( not <font color="Maroon" title="c17">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Maroon" title="c17">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E49">A46</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:31_2_3_1"/><span class="lab"><font color="Green" title="E50">A47</font></span>: 
<font color="Maroon" title="c17">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:31_2_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A48</font></span>: 
<font color="Maroon" title="c17">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:31_2"><span class="lab"><font color="Green" title="E42">A37</font></span></a>, <a class="txt" href="#E22:31_2"><span class="lab"><font color="Green" title="E44">A41</font></span></a>, <a class="txt" href="#E25:31_2"><span class="lab"><font color="Green" title="E46">A43</font></span></a>, <a class="txt" href="#E26:31_2"><span class="lab"><font color="Green" title="E47">A44</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">A2</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><span class="kw"> such that </span><br/><a NAME="E4:31_2_3_1"/><span class="lab"><font color="Green" title="E52">A49</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">A2</font>
 <span class="kw">and </span><br/><a NAME="E5:31_2_3_1"/><span class="lab"><font color="Green" title="E53">A50</font></span>: 
<font color="Maroon" title="c11">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">A2</font>
 <span class="kw">and </span><br/><a NAME="E6:31_2_3_1"/><span class="lab"><font color="Green" title="E54">A51</font></span>: 
<font color="Maroon" title="c12">c3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E9:31_2"><span class="lab"><font color="Green" title="E36">A31</font></span></a>, <a class="txt" href="#E11:31_2"><span class="lab"><font color="Green" title="E37">A32</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/><a NAME="E7:31_2_3_1"/>
 ex <font color="Olive" title="b1">A1</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A1</font> &amp; <font color="Maroon" title="c10">c1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A1</font> &amp; <font color="Maroon" title="c11">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E6:31_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="txt" href="#E9:31_2"><span class="lab"><font color="Green" title="E36">A31</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/><a NAME="E8:31_2_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E3:31_2"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="txt" href="#E6:31_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="txt" href="#E7:31_2"><span class="lab"><font color="Green" title="E35">A30</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/><a NAME="E9:31_2_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31_2"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="txt" href="#E9:31_2"><span class="lab"><font color="Green" title="E36">A31</font></span></a>, <a class="txt" href="#E4:31_2_3_1"><span class="lab"><font color="Green" title="E52">A49</font></span></a>, <a class="txt" href="#E5:31_2_3_1"><span class="lab"><font color="Green" title="E53">A50</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/><a NAME="E10:31_2_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E24:31_2"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="txt" href="#E11:31_2"><span class="lab"><font color="Green" title="E37">A32</font></span></a>, <a class="txt" href="#E1:31_2_3_1"><span class="lab"><font color="Green" title="E50">A47</font></span></a>, <a class="txt" href="#E4:31_2_3_1"><span class="lab"><font color="Green" title="E52">A49</font></span></a>, <a class="txt" href="#E6:31_2_3_1"><span class="lab"><font color="Green" title="E54">A51</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/><a NAME="E11:31_2_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E17:31_2"><span class="lab"><font color="Green" title="E41">A36</font></span></a>, <a class="txt" href="#E9:31_2"><span class="lab"><font color="Green" title="E36">A31</font></span></a>, <a class="txt" href="#E2:31_2_3_1"><span class="lab"><font color="Green" title="E51">A48</font></span></a>, <a class="ref" href="projred1.html#T26" target="_self" title="PROJRED1:th.26">Th26</a></span>;<br/><span class="kw">hence </span><a NAME="E12:31_2_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E6:31_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="txt" href="#E13:31_2"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E14:31_2"><span class="lab"><font color="Green" title="E39">A34</font></span></a>, <a class="ref" href="projred1.html#T26" target="_self" title="PROJRED1:th.26">Th26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E50">A52</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:31_2_3_2"/><span class="lab"><font color="Green" title="E51">A53</font></span>: 
<font color="Maroon" title="c17">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:31_2_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E3:31_2"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="txt" href="#E6:31_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="txt" href="#E7:31_2"><span class="lab"><font color="Green" title="E35">A30</font></span></a>, <a class="txt" href="#E24:31_2"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E3:31_2_3_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:31_2"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E14:31_2"><span class="lab"><font color="Green" title="E39">A34</font></span></a>, <a class="txt" href="#E22:31_2"><span class="lab"><font color="Green" title="E44">A41</font></span></a>, <a class="txt" href="#E25:31_2"><span class="lab"><font color="Green" title="E46">A43</font></span></a>, <a class="txt" href="#E20:31_2"><span class="lab"><font color="Green" title="E43">A38</font></span></a>, <a class="txt" href="#E1:31_2_3_2"><span class="lab"><font color="Green" title="E51">A53</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E3:31_2_3"/>
( <font color="Maroon" title="c17">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> or <font color="Maroon" title="c17">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">hence </span><a NAME="E4:31_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_2_3"><span class="lab"><font color="Green" title="E50">A52</font></span></a>, <a class="txt" href="#E1:31_2_3"><span class="lab"><font color="Green" title="E49">A46</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E28:31_2"/><span class="lab"><font color="Green" title="E49">A54</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> )</span><br/><div><span class="lab"><font color="Green" title="E50">A55</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31_2_4"/><span class="lab"><font color="Green" title="E51">A56</font></span>: 
<font color="Maroon" title="c6">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">R</font>
 <span class="kw">and </span><br/><a NAME="E2:31_2_4"/>
<font color="Maroon" title="c5">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> )</span><br/><a NAME="E3:31_2_4"/><span class="lab"><font color="Green" title="E52">A57</font></span>: 
<font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> or <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2_4_1"/><span class="lab"><font color="Green" title="E53">A58</font></span>: 
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">e</font> = <font color="Maroon" title="c18">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>;<br/>
<a NAME="E3:31_2_4_1"/><span class="lab"><font color="Green" title="E54">A59</font></span>: 
 ex <font color="Olive" title="b1">e9</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c19">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e9</font> &amp; <font color="Olive" title="b1">e9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E1:31_2_4_1"><span class="lab"><font color="Green" title="E53">A58</font></span></a></span>;<br/>
<a NAME="E4:31_2_4_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:31_2_4_1"/>
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31_2_4_1"><span class="lab"><font color="Green" title="E54">A59</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:31_2_4"/><span class="lab"><font color="Green" title="E53">A60</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2_4_2"/><span class="lab"><font color="Green" title="E54">A61</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span><br/>

<div><span class="lab"><font color="Green" title="E55">A62</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_4_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:31_2_4_2_1"/><span class="lab"><font color="Green" title="E56">A63</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span><br/><a NAME="E2:31_2_4_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_2_4"><span class="lab"><font color="Green" title="E52">A57</font></span></a>, <a class="txt" href="#E1:31_2_4_2"><span class="lab"><font color="Green" title="E54">A61</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>;<br/><a NAME="E3:31_2_4_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="projred1.html#T27" target="_self" title="PROJRED1:th.27">Th27</a></span>;<br/><a NAME="E4:31_2_4_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E7:31"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="ref" href="projred1.html#T27" target="_self" title="PROJRED1:th.27">Th27</a></span>;<br/><span class="kw">hence </span><a NAME="E5:31_2_4_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E7:31"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E27:31_2"><span class="lab"><font color="Green" title="E48">A45</font></span></a>, <a class="txt" href="#E1:31_2_4_2_1"><span class="lab"><font color="Green" title="E56">A63</font></span></a>, <a class="ref" href="projred1.html#T27" target="_self" title="PROJRED1:th.27">Th27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E56">A64</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_4_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31_2_4_2_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c1</font>
 <span class="kw">and </span><br/><a NAME="E2:31_2_4_2_2"/><span class="lab"><font color="Green" title="E57">A65</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font>
 <span class="kw">and </span><br/><a NAME="E3:31_2_4_2_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">a1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span><br/><a NAME="E4:31_2_4_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_4_2_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E58">A66</font></span>: 
<font color="Maroon" title="c14">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">a3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_4_2_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:31_2_4_2_2_1_1"/>
<font color="Maroon" title="c14">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_2_4_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A67</font></span>: 
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E7:31"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E17:31_2"><span class="lab"><font color="Green" title="E41">A36</font></span></a>, <a class="txt" href="#E19:31_2"><span class="lab"><font color="Green" title="E42">A37</font></span></a>, <a class="txt" href="#E1:31_2_4"><span class="lab"><font color="Green" title="E51">A56</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<a NAME="E3:31_2_4_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="projred1.html#T27" target="_self" title="PROJRED1:th.27">Th27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:31_2_4_2_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E13:31_2"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E15:31_2"><span class="lab"><font color="Green" title="E40">A35</font></span></a>, <a class="txt" href="#E2:31_2_4_2_2_1_1"><span class="lab"><font color="Green" title="E59">A67</font></span></a>, <a class="ref" href="projred1.html#T26" target="_self" title="PROJRED1:th.26">Th26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E59">A68</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_2_4"><span class="lab"><font color="Green" title="E52">A57</font></span></a>, <a class="txt" href="#E1:31_2_4_2"><span class="lab"><font color="Green" title="E54">A61</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>;<br/>
<a NAME="E3:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E60">A69</font></span>: 
<font color="Maroon" title="c14">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_4_2_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:31_2_4_2_2_1_2"/><span class="lab"><font color="Green" title="E61">A70</font></span>: 
<font color="Maroon" title="c14">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_2_4_2_2_1_2"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="projred1.html#T27" target="_self" title="PROJRED1:th.27">Th27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:31_2_4_2_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E13:31_2"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E15:31_2"><span class="lab"><font color="Green" title="E40">A35</font></span></a>, <a class="txt" href="#E1:31_2_4_2_2_1_2"><span class="lab"><font color="Green" title="E61">A70</font></span></a>, <a class="ref" href="projred1.html#T26" target="_self" title="PROJRED1:th.26">Th26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E61">A71</font></span>: 
<font color="Maroon" title="c15">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E17:31_2"><span class="lab"><font color="Green" title="E41">A36</font></span></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></span>;<br/>
<a NAME="E5:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E62">A72</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></span>;<br/>
<a NAME="E6:31_2_4_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A73</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31_2_4"><span class="lab"><font color="Green" title="E52">A57</font></span></a>, <a class="txt" href="#E1:31_2_4_2"><span class="lab"><font color="Green" title="E54">A61</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">y</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="projred1.html#T22" target="_self" title="PROJRED1:th.22">Th22</a></span>;<br/>
<a NAME="E8:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E64">A74</font></span>: 
<font color="Maroon" title="c19">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E6:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E63">A73</font></span></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">z</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="projred1.html#T22" target="_self" title="PROJRED1:th.22">Th22</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c21">B3</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><span class="kw"> such that </span><br/><a NAME="E11:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E65">A75</font></span>: 
( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">B3</font> &amp; <font color="Maroon" title="c18">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">B3</font> &amp; <font color="Maroon" title="c19">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">B3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E6:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E63">A73</font></span></a>, <a class="txt" href="#E8:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E64">A74</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<a NAME="E12:31_2_4_2_2_1"/>
<font color="Maroon" title="c18">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31_2_4"><span class="lab"><font color="Green" title="E52">A57</font></span></a>, <a class="txt" href="#E1:31_2_4_2"><span class="lab"><font color="Green" title="E54">A61</font></span></a>, <a class="txt" href="#E5:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E62">A72</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<a NAME="E13:31_2_4_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A76</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E2:31_2_4_2_2"><span class="lab"><font color="Green" title="E57">A65</font></span></a>, <a class="txt" href="#E11:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E65">A75</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">A1</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><span class="kw"> such that </span><br/><a NAME="E15:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E67">A77</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c22">A1</font>
 <span class="kw">and </span><br/><a NAME="E16:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E68">A78</font></span>: 
( <font color="Maroon" title="c14">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c22">A1</font> &amp; <font color="Maroon" title="c15">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c22">A1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E17:31_2"><span class="lab"><font color="Green" title="E41">A36</font></span></a>, <a class="txt" href="#E19:31_2"><span class="lab"><font color="Green" title="E42">A37</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<a NAME="E17:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E69">A79</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c14">a2</font>,<font color="Maroon" title="c15">a3</font>,<font color="Maroon" title="c3">q</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c22">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E67">A77</font></span></a>, <a class="txt" href="#E16:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E68">A78</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E18:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E70">A80</font></span>: 
<font color="Maroon" title="c20">z</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E8:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E64">A74</font></span></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">B1</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><span class="kw"> such that </span><br/><a NAME="E20:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E71">A81</font></span>: 
( <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c23">B1</font> &amp; <font color="Maroon" title="c19">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c23">B1</font> &amp; <font color="Maroon" title="c20">z</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c23">B1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E8:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E64">A74</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<a NAME="E21:31_2_4_2_2_1"/>
<font color="Maroon" title="c18">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31_2_4"><span class="lab"><font color="Green" title="E52">A57</font></span></a>, <a class="txt" href="#E1:31_2_4_2"><span class="lab"><font color="Green" title="E54">A61</font></span></a>, <a class="txt" href="#E5:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E62">A72</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<a NAME="E22:31_2_4_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A82</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c13">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E13:31_2"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c24">A3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <font color="Maroon" title="c1">IPP</font><span class="kw"> such that </span><br/><a NAME="E24:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E73">A83</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c24">A3</font>
 <span class="kw">and </span><br/><a NAME="E25:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E74">A84</font></span>: 
<font color="Maroon" title="c13">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c24">A3</font>
 <span class="kw">and </span><br/><a NAME="E26:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E75">A85</font></span>: 
<font color="Maroon" title="c14">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c24">A3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E13:31_2"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E17:31_2"><span class="lab"><font color="Green" title="E41">A36</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<a NAME="E27:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E76">A86</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c14">a2</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c13">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c24">A3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E73">A83</font></span></a>, <a class="txt" href="#E25:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E74">A84</font></span></a>, <a class="txt" href="#E26:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E75">A85</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E28:31_2_4_2_2_1"/>
<font color="Maroon" title="c22">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">A3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_4_2_2_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:31_2_4_2_2_1_3"/>
<font color="Maroon" title="c22">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">A3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_2_4_2_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">A3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:31_2"><span class="lab"><font color="Green" title="E49">A54</font></span></a>, <a class="txt" href="#E3:31_2"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="txt" href="#E4:31_2"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="txt" href="#E15:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E67">A77</font></span></a>, <a class="txt" href="#E24:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E73">A83</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:31_2_4_2_2_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E6:31_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="txt" href="#E7:31_2"><span class="lab"><font color="Green" title="E35">A30</font></span></a>, <a class="txt" href="#E13:31_2"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E14:31_2"><span class="lab"><font color="Green" title="E39">A34</font></span></a>, <a class="txt" href="#E24:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E73">A83</font></span></a>, <a class="txt" href="#E25:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E74">A84</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E29:31_2_4_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A87</font></span>: 
<font color="Maroon" title="c22">A1</font>,<font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c24">A3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E15:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E67">A77</font></span></a>, <a class="txt" href="#E24:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E73">A83</font></span></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></span>;<br/>
<a NAME="E30:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E78">A88</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c14">a2</font>,<font color="Maroon" title="c19">y</font>,<font color="Maroon" title="c4">c</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c6">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E17:31_2"><span class="lab"><font color="Green" title="E41">A36</font></span></a>, <a class="txt" href="#E8:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E64">A74</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E31:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E79">A89</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c19">y</font>,<font color="Maroon" title="c18">x</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c21">B3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E65">A75</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E32:31_2_4_2_2_1"/>
<font color="Maroon" title="c20">z</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E8:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E64">A74</font></span></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></span>;<br/>
<a NAME="E33:31_2_4_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A90</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c15">a3</font>,<font color="Maroon" title="c20">z</font>,<font color="Maroon" title="c4">c</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c7">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E4:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E61">A71</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E34:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E81">A91</font></span>: 
( <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c3">q</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c9">A</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c15">a3</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c13">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c16">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_2"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="txt" href="#E4:31_2"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="txt" href="#E22:31_2"><span class="lab"><font color="Green" title="E44">A41</font></span></a>, <a class="txt" href="#E24:31_2"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="txt" href="#E25:31_2"><span class="lab"><font color="Green" title="E46">A43</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E35:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E82">A92</font></span>: 
<font color="Maroon" title="c14">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E13:31_2"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></span>;<br/>
<a NAME="E36:31_2_4_2_2_1"/>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c19">y</font>,<font color="Maroon" title="c20">z</font>,<font color="Maroon" title="c3">q</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c23">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E71">A81</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">O</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><span class="kw"> such that </span><br/><a NAME="E38:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E83">A93</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c20">z</font>,<font color="Maroon" title="c18">x</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c25">O</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E60">A69</font></span></a>, <a class="txt" href="#E1:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E58">A66</font></span></a>, <a class="txt" href="#E13:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E66">A76</font></span></a>, <a class="txt" href="#E30:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E78">A88</font></span></a>, <a class="txt" href="#E17:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E69">A79</font></span></a>, <a class="txt" href="#E27:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E76">A86</font></span></a>, <a class="txt" href="#E31:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E79">A89</font></span></a>, <a class="txt" href="#E34:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E81">A91</font></span></a>, <a class="txt" href="#E22:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E72">A82</font></span></a>, <a class="txt" href="#E33:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E80">A90</font></span></a>, <a class="txt" href="#E29:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E77">A87</font></span></a>, <a class="txt" href="#E35:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E82">A92</font></span></a>, <a class="ref" href="projred1.html#T14" target="_self" title="PROJRED1:th.14">Th14</a></span>;<br/>
<a NAME="E39:31_2_4_2_2_1"/><span class="lab"><font color="Green" title="E84">A94</font></span>: 
<font color="Maroon" title="c17">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E83">A93</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E40:31_2_4_2_2_1"/>
( <font color="Maroon" title="c18">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">O</font> &amp; <font color="Maroon" title="c20">z</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">O</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E83">A93</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E41:31_2_4_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:31_2"><span class="lab"><font color="Green" title="E48">A45</font></span></a>, <a class="txt" href="#E6:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E63">A73</font></span></a>, <a class="txt" href="#E18:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E70">A80</font></span></a>, <a class="txt" href="#E39:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E84">A94</font></span></a>, <a class="txt" href="#E2:31_2_4_2_2_1"><span class="lab"><font color="Green" title="E59">A68</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:31_2_4_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E57">A95</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_4_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:31_2_4_2_3"/><span class="lab"><font color="Green" title="E58">A96</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span><br/><a NAME="E2:31_2_4_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_4_2_3_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:31_2_4_2_3_1"/><span class="lab"><font color="Green" title="E59">A97</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_2_4"><span class="lab"><font color="Green" title="E52">A57</font></span></a>, <a class="txt" href="#E1:31_2_4_2"><span class="lab"><font color="Green" title="E54">A61</font></span></a>, <a class="txt" href="#E1:31_2_4_2_3"><span class="lab"><font color="Green" title="E58">A96</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">A2</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><span class="kw"> such that </span><br/><a NAME="E3:31_2_4_2_3_1"/><span class="lab"><font color="Green" title="E60">A98</font></span>: 
( <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">A2</font> &amp; <font color="Maroon" title="c11">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">A2</font> )
 <span class="kw">and </span><br/><a NAME="E4:31_2_4_2_3_1"/><span class="lab"><font color="Green" title="E61">A99</font></span>: 
<font color="Maroon" title="c12">c3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E9:31_2"><span class="lab"><font color="Green" title="E36">A31</font></span></a>, <a class="txt" href="#E11:31_2"><span class="lab"><font color="Green" title="E37">A32</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<a NAME="E5:31_2_4_2_3_1"/>
 ex <font color="Olive" title="b1">A1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A1</font> &amp; <font color="Maroon" title="c10">c1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A1</font> &amp; <font color="Maroon" title="c11">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E6:31_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="txt" href="#E9:31_2"><span class="lab"><font color="Green" title="E36">A31</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<a NAME="E6:31_2_4_2_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E3:31_2"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="txt" href="#E6:31_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="txt" href="#E7:31_2"><span class="lab"><font color="Green" title="E35">A30</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<a NAME="E7:31_2_4_2_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31_2"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="txt" href="#E9:31_2"><span class="lab"><font color="Green" title="E36">A31</font></span></a>, <a class="txt" href="#E3:31_2_4_2_3_1"><span class="lab"><font color="Green" title="E60">A98</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:31_2_4_2_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="txt" href="#E7:31_2"><span class="lab"><font color="Green" title="E35">A30</font></span></a>, <a class="txt" href="#E24:31_2"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="txt" href="#E11:31_2"><span class="lab"><font color="Green" title="E37">A32</font></span></a>, <a class="txt" href="#E27:31_2"><span class="lab"><font color="Green" title="E48">A45</font></span></a>, <a class="txt" href="#E1:31_2_4_2_3"><span class="lab"><font color="Green" title="E58">A96</font></span></a>, <a class="txt" href="#E1:31_2_4_2_3_1"><span class="lab"><font color="Green" title="E59">A97</font></span></a>, <a class="txt" href="#E4:31_2_4_2_3_1"><span class="lab"><font color="Green" title="E61">A99</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:31_2_4_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_4_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:31_2_4_2_4"/><span class="lab"><font color="Green" title="E58">A100</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span><br/><a NAME="E2:31_2_4_2_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_2_4"><span class="lab"><font color="Green" title="E52">A57</font></span></a>, <a class="txt" href="#E1:31_2_4_2"><span class="lab"><font color="Green" title="E54">A61</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>;<br/><span class="kw">hence </span><a NAME="E3:31_2_4_2_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:31_2"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E19:31_2"><span class="lab"><font color="Green" title="E42">A37</font></span></a>, <a class="txt" href="#E22:31_2"><span class="lab"><font color="Green" title="E44">A41</font></span></a>, <a class="txt" href="#E25:31_2"><span class="lab"><font color="Green" title="E46">A43</font></span></a>, <a class="txt" href="#E27:31_2"><span class="lab"><font color="Green" title="E48">A45</font></span></a>, <a class="txt" href="#E1:31_2_4_2_4"><span class="lab"><font color="Green" title="E58">A100</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:31_2_4_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_2_4_2"><span class="lab"><font color="Green" title="E55">A62</font></span></a>, <a class="txt" href="#E4:31_2_4_2"><span class="lab"><font color="Green" title="E57">A95</font></span></a>, <a class="txt" href="#E3:31_2_4_2"><span class="lab"><font color="Green" title="E56">A64</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:31_2_4"/><span class="lab"><font color="Green" title="E54">A101</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_4_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> or <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2_4_3"/><span class="lab"><font color="Green" title="E55">A102</font></span>: 
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">e</font> = <font color="Maroon" title="c18">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> ;<br/>
<a NAME="E3:31_2_4_3"/>
<font color="Maroon" title="c19">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:31_2"><span class="lab"><font color="Green" title="E48">A45</font></span></a>, <a class="txt" href="#E1:31_2_4_3"><span class="lab"><font color="Green" title="E55">A102</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:31_2_4_3"/>
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:31_2_4"/>
<font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_4_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> or <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2_4_4"/><span class="lab"><font color="Green" title="E55">A103</font></span>: 
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">e</font> = <font color="Maroon" title="c18">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>;<br/>
<a NAME="E3:31_2_4_4"/>
 ex <font color="Olive" title="b1">e9</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c19">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e9</font> &amp; <font color="Olive" title="b1">e9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E1:31_2_4_4"><span class="lab"><font color="Green" title="E55">A103</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:31_2_4_4"/>
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:31_2"><span class="lab"><font color="Green" title="E48">A45</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:31_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A104</font></span>: 
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c17">o</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_2_4"><span class="lab"><font color="Green" title="E54">A101</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><a NAME="E8:31_2_4"/><span class="lab"><font color="Green" title="E56">A105</font></span>: 
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>,the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:31"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/><a NAME="E9:31_2_4"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_4_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> or <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2_4_5"/>
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font></span><br/>

<a NAME="E2:31_2_4_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A106</font></span>: 
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">e</font> = <font color="Maroon" title="c18">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> ;<br/>
<a NAME="E4:31_2_4_5"/>
<font color="Maroon" title="c19">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E2:31_2_4_5"><span class="lab"><font color="Green" title="E57">A106</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:31_2_4_5"/>
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E10:31_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_2_4"><span class="lab"><font color="Green" title="E52">A57</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E11:31_2_4"/>
 ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:31_2"><span class="lab"><font color="Green" title="E48">A45</font></span></a>, <a class="txt" href="#E4:31_2_4"><span class="lab"><font color="Green" title="E53">A60</font></span></a>, <a class="txt" href="#E7:31_2_4"><span class="lab"><font color="Green" title="E55">A104</font></span></a>, <a class="txt" href="#E8:31_2_4"><span class="lab"><font color="Green" title="E56">A105</font></span></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_5"><span class="kw">now </span></a><div class="add"><a NAME="E1:31_2_5"/><span class="lab"><font color="Green" title="E51">A107</font></span>: 
<font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> or <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2_5_1"/><span class="lab"><font color="Green" title="E52">A108</font></span>: 
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">e</font> = <font color="Maroon" title="c18">e</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>;<br/>
<a NAME="E3:31_2_5_1"/><span class="lab"><font color="Green" title="E53">A109</font></span>: 
 ex <font color="Olive" title="b1">e9</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c19">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e9</font> &amp; <font color="Olive" title="b1">e9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E1:31_2_5_1"><span class="lab"><font color="Green" title="E52">A108</font></span></a></span>;<br/>
<a NAME="E4:31_2_5_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:31_2_5_1"/>
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31_2_5_1"><span class="lab"><font color="Green" title="E53">A109</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E2:31_2_5"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> or <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2_5_2"/>
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font></span><br/>

<a NAME="E2:31_2_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A110</font></span>: 
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">e</font> = <font color="Maroon" title="c18">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> ;<br/>
<a NAME="E4:31_2_5_2"/>
<font color="Maroon" title="c19">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E2:31_2_5_2"><span class="lab"><font color="Green" title="E52">A110</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:31_2_5_2"/>
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:31_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A111</font></span>: 
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_2_5"><span class="lab"><font color="Green" title="E51">A107</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><a NAME="E4:31_2_5"/><span class="lab"><font color="Green" title="E53">A112</font></span>: 
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>,the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:31"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/><span class="kw">assume </span><a NAME="E5:31_2_5"/><span class="lab"><font color="Green" title="E54">A113</font></span>: 
<font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b2">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b2">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b2">o</font>,<font color="Maroon" title="c7">R</font> )</span><br/><a NAME="E6:31_2_5"/><span class="lab"><font color="Green" title="E55">A114</font></span>: 
<font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_5_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> or <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2_5_3"/><span class="lab"><font color="Green" title="E56">A115</font></span>: 
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">e</font> = <font color="Maroon" title="c18">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>;<br/>
<a NAME="E3:31_2_5_3"/>
 ex <font color="Olive" title="b1">e9</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c19">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e9</font> &amp; <font color="Olive" title="b1">e9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E1:31_2_5_3"><span class="lab"><font color="Green" title="E56">A115</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:31_2_5_3"/>
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E5:31_2_5"><span class="lab"><font color="Green" title="E54">A113</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:31_2_5"/><span class="lab"><font color="Green" title="E56">A116</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_5_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2_5_4"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span><br/>

<a NAME="E2:31_2_5_4"/><span class="kw">then </span>
( <font color="Maroon" title="c18">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E5:31_2_5"><span class="lab"><font color="Green" title="E54">A113</font></span></a>, <a class="txt" href="#E1:31_2_5"><span class="lab"><font color="Green" title="E51">A107</font></span></a>, <a class="txt" href="#E6:31_2_5"><span class="lab"><font color="Green" title="E55">A114</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:31_2_5_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E5:31_2_5"><span class="lab"><font color="Green" title="E54">A113</font></span></a>, <a class="ref" href="projred1.html#T21" target="_self" title="PROJRED1:th.21">Th21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">take </span><font color="Maroon" title="c18">q</font> = <font color="Maroon" title="c3">q</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> )</span><br/><a NAME="E8:31_2_5"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_5_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> or <font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2_5_5"/><span class="lab"><font color="Green" title="E57">A117</font></span>: 
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">e</font> = <font color="Maroon" title="c19">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> ;<br/>
<a NAME="E3:31_2_5_5"/>
<font color="Maroon" title="c20">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E5:31_2_5"><span class="lab"><font color="Green" title="E54">A113</font></span></a>, <a class="txt" href="#E1:31_2_5_5"><span class="lab"><font color="Green" title="E57">A117</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:31_2_5_5"/>
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E9:31_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31_2_5"><span class="lab"><font color="Green" title="E55">A114</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E10:31_2_5"/>
 ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E5:31_2_5"><span class="lab"><font color="Green" title="E54">A113</font></span></a>, <a class="txt" href="#E3:31_2_5"><span class="lab"><font color="Green" title="E52">A111</font></span></a>, <a class="txt" href="#E7:31_2_5"><span class="lab"><font color="Green" title="E56">A116</font></span></a>, <a class="txt" href="#E4:31_2_5"><span class="lab"><font color="Green" title="E53">A112</font></span></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E31:31_2"/>
 ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_2"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E29:31_2"><span class="lab"><font color="Green" title="E50">A55</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><span class="kw">now </span></a><div class="add"><a NAME="E1:31_3"/><span class="lab"><font color="Green" title="E31">A118</font></span>: 
<font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> or <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_3_1"/><span class="lab"><font color="Green" title="E32">A119</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">e</font> = <font color="Maroon" title="c9">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>;<br/>
<a NAME="E3:31_3_1"/><span class="lab"><font color="Green" title="E33">A120</font></span>: 
 ex <font color="Olive" title="b1">e9</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e9</font> &amp; <font color="Olive" title="b1">e9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E1:31_3_1"><span class="lab"><font color="Green" title="E32">A119</font></span></a></span>;<br/>
<a NAME="E4:31_3_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:31_3_1"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31_3_1"><span class="lab"><font color="Green" title="E33">A120</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:31_3"/><span class="lab"><font color="Green" title="E32">A121</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> )</span><br/><a NAME="E3:31_3"/><span class="lab"><font color="Green" title="E33">A122</font></span>: 
<font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> or <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_3_2"/><span class="lab"><font color="Green" title="E34">A123</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">e</font> = <font color="Maroon" title="c9">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>;<br/>
<a NAME="E3:31_3_2"/>
 ex <font color="Olive" title="b1">e9</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e9</font> &amp; <font color="Olive" title="b1">e9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E1:31_3_2"><span class="lab"><font color="Green" title="E34">A123</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:31_3_2"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E2:31_3"><span class="lab"><font color="Green" title="E32">A121</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:31_3"/><span class="lab"><font color="Green" title="E34">A124</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_3_3"/><span class="lab"><font color="Green" title="E35">A125</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span><br/>

<a NAME="E2:31_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A126</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_3"><span class="lab"><font color="Green" title="E32">A121</font></span></a>, <a class="txt" href="#E1:31_3"><span class="lab"><font color="Green" title="E31">A118</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>;<br/>
<a NAME="E3:31_3_3"/><span class="lab"><font color="Green" title="E37">A127</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E2:31_3"><span class="lab"><font color="Green" title="E32">A121</font></span></a>, <a class="txt" href="#E3:31_3"><span class="lab"><font color="Green" title="E33">A122</font></span></a>, <a class="txt" href="#E1:31_3_3"><span class="lab"><font color="Green" title="E35">A125</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">y</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="projred1.html#T22" target="_self" title="PROJRED1:th.22">Th22</a></span>;<br/>
<a NAME="E5:31_3_3"/><span class="lab"><font color="Green" title="E38">A128</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31_3_3"><span class="lab"><font color="Green" title="E37">A127</font></span></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">z</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E2:31_3"><span class="lab"><font color="Green" title="E32">A121</font></span></a>, <a class="ref" href="projred1.html#T22" target="_self" title="PROJRED1:th.22">Th22</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">A</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><span class="kw"> such that </span><br/><a NAME="E8:31_3_3"/><span class="lab"><font color="Green" title="E39">A129</font></span>: 
( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">A</font> &amp; <font color="Maroon" title="c10">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></span>;<br/>
<a NAME="E9:31_3_3"/><span class="lab"><font color="Green" title="E40">A130</font></span>: 
<font color="Maroon" title="c11">z</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E2:31_3"><span class="lab"><font color="Green" title="E32">A121</font></span></a>, <a class="txt" href="#E5:31_3_3"><span class="lab"><font color="Green" title="E38">A128</font></span></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">C</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><span class="kw"> such that </span><br/><a NAME="E11:31_3_3"/><span class="lab"><font color="Green" title="E41">A131</font></span>: 
( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">C</font> &amp; <font color="Maroon" title="c10">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">C</font> )
 <span class="kw">and </span><br/><a NAME="E12:31_3_3"/><span class="lab"><font color="Green" title="E42">A132</font></span>: 
<font color="Maroon" title="c11">z</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E2:31_3"><span class="lab"><font color="Green" title="E32">A121</font></span></a>, <a class="txt" href="#E5:31_3_3"><span class="lab"><font color="Green" title="E38">A128</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<a NAME="E13:31_3_3"/><span class="lab"><font color="Green" title="E43">A133</font></span>: 
<font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E5:31_3_3"><span class="lab"><font color="Green" title="E38">A128</font></span></a>, <a class="txt" href="#E8:31_3_3"><span class="lab"><font color="Green" title="E39">A129</font></span></a>, <a class="txt" href="#E11:31_3_3"><span class="lab"><font color="Green" title="E41">A131</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">B</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><span class="kw"> such that </span><br/><a NAME="E15:31_3_3"/><span class="lab"><font color="Green" title="E44">A134</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">B</font>
 <span class="kw">and </span><br/><a NAME="E16:31_3_3"/><span class="lab"><font color="Green" title="E45">A135</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">B</font>
 <span class="kw">and </span><br/><a NAME="E17:31_3_3"/><span class="lab"><font color="Green" title="E46">A136</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31_3_3"><span class="lab"><font color="Green" title="E37">A127</font></span></a>, <a class="txt" href="#E5:31_3_3"><span class="lab"><font color="Green" title="E38">A128</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<a NAME="E18:31_3_3"/>
<font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E5:31_3_3"><span class="lab"><font color="Green" title="E38">A128</font></span></a>, <a class="txt" href="#E8:31_3_3"><span class="lab"><font color="Green" title="E39">A129</font></span></a>, <a class="txt" href="#E15:31_3_3"><span class="lab"><font color="Green" title="E44">A134</font></span></a>, <a class="txt" href="#E17:31_3_3"><span class="lab"><font color="Green" title="E46">A136</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:31_3_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E2:31_3"><span class="lab"><font color="Green" title="E32">A121</font></span></a>, <a class="txt" href="#E3:31_3_3"><span class="lab"><font color="Green" title="E37">A127</font></span></a>, <a class="txt" href="#E2:31_3_3"><span class="lab"><font color="Green" title="E36">A126</font></span></a>, <a class="txt" href="#E15:31_3_3"><span class="lab"><font color="Green" title="E44">A134</font></span></a>, <a class="txt" href="#E16:31_3_3"><span class="lab"><font color="Green" title="E45">A135</font></span></a>, <a class="txt" href="#E9:31_3_3"><span class="lab"><font color="Green" title="E40">A130</font></span></a>, <a class="txt" href="#E12:31_3_3"><span class="lab"><font color="Green" title="E42">A132</font></span></a>, <a class="txt" href="#E13:31_3_3"><span class="lab"><font color="Green" title="E43">A133</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:31_3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span> or <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_3_4"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font></span><br/>

<a NAME="E2:31_3_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A137</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="projred1.html#T25" target="_self" title="PROJRED1:th.25">Th25</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">e</font> = <font color="Maroon" title="c9">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> ;<br/>
<a NAME="E4:31_3_4"/>
<font color="Maroon" title="c10">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E2:31_3_4"><span class="lab"><font color="Green" title="E35">A137</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:31_3_4"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:31_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A138</font></span>: 
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_3"><span class="lab"><font color="Green" title="E31">A118</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><a NAME="E7:31_3"/><span class="lab"><font color="Green" title="E36">A139</font></span>: 
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>,the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:31"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/><a NAME="E8:31_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span> or <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_3_5"/><span class="lab"><font color="Green" title="E37">A140</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">e</font> = <font color="Maroon" title="c9">e</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> ;<br/>
<a NAME="E3:31_3_5"/>
<font color="Maroon" title="c10">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E2:31_3"><span class="lab"><font color="Green" title="E32">A121</font></span></a>, <a class="txt" href="#E1:31_3_5"><span class="lab"><font color="Green" title="E37">A140</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:31_3_5"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E9:31_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_3"><span class="lab"><font color="Green" title="E33">A122</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E10:31_3"/>
 ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E2:31_3"><span class="lab"><font color="Green" title="E32">A121</font></span></a>, <a class="txt" href="#E6:31_3"><span class="lab"><font color="Green" title="E35">A138</font></span></a>, <a class="txt" href="#E4:31_3"><span class="lab"><font color="Green" title="E34">A124</font></span></a>, <a class="txt" href="#E7:31_3"><span class="lab"><font color="Green" title="E36">A139</font></span></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E14:31"/>
 ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">R</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">R</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:31"><span class="lab"><font color="Green" title="E30">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
