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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">a</font>, <font color="Olive" title="b2">b</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="b3">A</font>, <font color="Olive" title="b4">B</font>, <font color="Olive" title="b5">C</font>, <font color="Olive" title="b6">Q</font>, <font color="Olive" title="b7">O</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">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">A</font> &amp; not <font color="Olive" title="b2">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">B</font> &amp; not <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b5">C</font> &amp; not <font color="Olive" title="b2">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b5">C</font> &amp; not <font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font>,<font color="Olive" title="b5">C</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; <font color="Olive" title="b4">B</font>,<font color="Olive" title="b5">C</font>,<font color="Olive" title="b6">Q</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; not <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b6">Q</font> &amp; <font color="Olive" title="b4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">Q</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b7">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b7">O</font> holds <br/> ex <font color="Olive" title="b8">q</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="Olive" title="b8">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b7">O</font> &amp; not <font color="Olive" title="b8">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">B</font> &amp; not <font color="Olive" title="b8">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b6">Q</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Olive" title="b5">C</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">B</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="b3">A</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">C</font></span>)</span> <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="Olive" title="b6">Q</font>,<font color="Olive" title="b8">q</font>,<font color="Olive" title="b4">B</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="b3">A</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b6">Q</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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">A</font>, <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">C</font>, <font color="Olive" title="b4">Q</font>, <font color="Olive" title="b5">O</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">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">B</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">C</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">C</font> &amp; not <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">C</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; <font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">C</font>,<font color="Olive" title="b4">Q</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; not <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">Q</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">Q</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b5">O</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b5">O</font> holds <br/> ex <font color="Olive" title="b6">q</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="Olive" title="b6">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b5">O</font> &amp; not <font color="Olive" title="b6">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">B</font> &amp; not <font color="Olive" title="b6">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">Q</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Olive" title="b3">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Olive" title="b2">B</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">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b3">C</font></span>)</span> <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="Olive" title="b4">Q</font>,<font color="Olive" title="b6">q</font>,<font color="Olive" title="b2">B</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">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b4">Q</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font>, <font color="Maroon" title="c6">C</font>, <font color="Maroon" title="c7">Q</font>, <font color="Maroon" title="c8">O</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">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">A</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">B</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">C</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">C</font> &amp; not <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">C</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c7">Q</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; not <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">Q</font> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">Q</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">O</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">O</font> implies  ex <font color="Olive" title="b1">q</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="Olive" title="b1">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">O</font> &amp; not <font color="Olive" title="b1">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">B</font> &amp; not <font color="Olive" title="b1">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">Q</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">B</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="c4">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">C</font></span>)</span> <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">Q</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c5">B</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="c4">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">Q</font></span>)</span> ) )</span><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:26"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
not <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E2:26"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">and </span><br/><a NAME="E3:26"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
not <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">C</font>
 <span class="kw">and </span><br/><a NAME="E4:26"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">C</font>
 <span class="kw">and </span><br/><a NAME="E5:26"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
not <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">C</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a> 
 <span class="kw">and </span><br/><a NAME="E6:26"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c7">Q</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a> 
 <span class="kw">and </span><br/><a NAME="E7:26"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
not <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">Q</font>
 <span class="kw">and </span><br/><a NAME="E8:26"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
( <font color="Maroon" title="c5">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">Q</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">O</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">O</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> 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="Olive" title="b1">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">O</font> &amp; not <font color="Olive" title="b1">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">B</font> &amp; not <font color="Olive" title="b1">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">Q</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">B</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="c4">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">C</font></span>)</span> <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">Q</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c5">B</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="c4">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">Q</font></span>)</span> )</span><br/>

<a NAME="E9:26"/><span class="lab"><font color="Green" title="E27">A9</font></span>: 
(  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">C</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="projred2.html#T8" target="_self" title="PROJRED2:th.8">Th8</a></span>;<br/>
<a NAME="E10:26"/>
not <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c6">C</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:26_1"/>
<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c6">C</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a> 
 ; <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="E2:26_1"/><span class="kw">then </span>
 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/>( <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">B</font> &amp; <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:26_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2: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">then </span><span class="kw">consider </span><font color="Maroon" title="c9">q</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="E12:26"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">O</font>
 <span class="kw">and </span><br/><a NAME="E13:26"/><span class="lab"><font color="Green" title="E29">A11</font></span>: 
( not <font color="Maroon" title="c9">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">B</font> &amp; not <font color="Maroon" title="c9">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">Q</font> )
 <span class="kw">and </span><br/><a NAME="E14:26"/><span class="lab"><font color="Green" title="E30">A12</font></span>: 
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</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">B</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">C</font></span>)</span> <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">Q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</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">B</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c7">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="projred2.html#T21" target="_self" title="PROJRED2:th.21">Th21</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">q</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">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">O</font> &amp; not <font color="Maroon" title="c9">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">B</font> &amp; not <font color="Maroon" title="c9">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">Q</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">B</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="c4">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">C</font></span>)</span> <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">Q</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c5">B</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="c4">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">Q</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E15:26"/>
( <font color="Maroon" title="c9">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">O</font> &amp; not <font color="Maroon" title="c9">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">B</font> &amp; not <font color="Maroon" title="c9">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">Q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E29">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"> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">B</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="c4">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">C</font></span>)</span> <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">Q</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c5">B</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="c4">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">Q</font></span>)</span></span><br/>

<a NAME="E16:26"/><span class="lab"><font color="Green" title="E31">A13</font></span>: 
 <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c7">Q</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="ref" href="projred2.html#T8" target="_self" title="PROJRED2:th.8">Th8</a></span>;<br/>
<a NAME="E17:26"/><span class="lab"><font color="Green" title="E32">A14</font></span>: 
 <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c7">Q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="projred2.html#T8" target="_self" title="PROJRED2:th.8">Th8</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">B</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="c4">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">C</font></span>)</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="c5">B</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">C</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </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="c4">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">C</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="projred2.html#T9" target="_self" title="PROJRED2:th.9">Th9</a></span>
<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="c5">B</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">C</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</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">C</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="projred2.html#T9" target="_self" title="PROJRED2:th.9">Th9</a></span>
<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="c7">Q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</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">B</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c7">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="funct_1.html#T66" title="FUNCT_1:th.66">FUNCT_1:66</a></span>
<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="c5">B</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c7">Q</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</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="c7">Q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E17:26"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="ref" href="funct_1.html#T66" title="FUNCT_1:th.66">FUNCT_1:66</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c7">Q</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</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="c7">Q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="ref" href="projred2.html#T9" target="_self" title="PROJRED2:th.9">Th9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c7">Q</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c5">B</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="c4">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">Q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="projred2.html#T9" target="_self" title="PROJRED2:th.9">Th9</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>
