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<b>let </b><font color="Maroon" title="c1">IPP</font> be  <a href="incproj.html#V6" title="INCPROJ:attr.6">2-dimensional</a> <a href="incproj.html#V10" title="INCPROJ:attr.10">Desarguesian</a> <a href="incproj.html#NM2" title="INCPROJ:NM.2">IncProjSp</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">q</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>;<br/><b>let </b><font color="Maroon" title="c6">A</font>, <font color="Maroon" title="c7">B</font>, <font color="Maroon" title="c8">C</font>, <font color="Maroon" title="c9">O</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font>;<br/>

















<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E20">A1</font></i>: 
( 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">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="c7">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="c8">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="c8">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="Maroon" title="c6">A</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="c8">C</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="c9">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="c9">O</font> &amp; <font color="Maroon" title="c5">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">O</font> &amp; not <font color="Maroon" title="c5">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; not <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c8">C</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; not <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c8">C</font>,<font color="Maroon" title="c9">O</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  )
 ;<br/>

<b>consider </b><font color="Maroon" title="c10">d</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E3:27"/><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c10">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">C</font> &amp; <font color="Maroon" title="c10">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">B</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14" title="INCPROJ:def.14">INCPROJ:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">O1</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E5:27"/><i><font color="Green" title="E22">A3</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">O1</font> &amp; <font color="Maroon" title="c10">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">O1</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">O3</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E7:27"/><i><font color="Green" title="E23">A4</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">O3</font> &amp; <font color="Maroon" title="c10">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">O3</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">p</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E9:27"/><i><font color="Green" title="E24">A5</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c13">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">O1</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14" title="INCPROJ:def.14">INCPROJ:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">O2</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E11:27"/><i><font color="Green" title="E25">A6</font></i>: 
( <font color="Maroon" title="c5">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">O2</font> &amp; <font color="Maroon" title="c13">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">O2</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">pp'</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E13:27"/><i><font color="Green" title="E26">A7</font></i>: 
( <font color="Maroon" title="c15">pp'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">O3</font> &amp; <font color="Maroon" title="c15">pp'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">O2</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14" title="INCPROJ:def.14">INCPROJ:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">Q</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E15:27"/><i><font color="Green" title="E27">A8</font></i>: 
( <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">Q</font> &amp; <font color="Maroon" title="c15">pp'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">Q</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1"/><i><font color="Green" title="E28">A9</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 ;<br/><a NAME="E2:27_1"/><b>then </b>
( 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">A</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="c8">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="c7">B</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="c8">C</font> &amp; not <font color="Maroon" title="c5">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">A</font> &amp; not <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c8">C</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; not <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c8">C</font>,<font color="Maroon" title="c9">O</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &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="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c13">p</font> <a href="incproj.html#R3" title="INCPROJ:pred.3">on</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c10">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c10">d</font> <a href="incproj.html#R3" title="INCPROJ:pred.3">on</a> <font color="Maroon" title="c8">C</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">q</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c9">O</font> &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c15">pp'</font> <a href="incproj.html#R3" title="INCPROJ:pred.3">on</a> <font color="Maroon" title="c16">Q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c10">d</font>,<font color="Maroon" title="c13">p</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c11">O1</font> &amp; <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">pp'</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c14">O2</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c10">d</font>,<font color="Maroon" title="c15">pp'</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c12">O3</font> )
 <b>by </b><i><a class="txt" href="projred2.html#E1:27"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred2.html#E3:27"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="projred2.html#E5:27"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="projred2.html#E7:27"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="projred2.html#E9:27"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="projred2.html#E11:27"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="projred2.html#E13:27"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="projred2.html#E15:27"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="incproj.html#D7" title="INCPROJ:def.7">INCPROJ:def 7</a>, <a class="ref" href="incproj.html#D8" title="INCPROJ:def.8">INCPROJ:def 8</a></i>;<br/><a NAME="E3:27_1"/><b>then </b><i><font color="Green" title="E29">A10</font></i>: 
( <font color="Maroon" title="c16">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c16">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">C</font> &amp; not <font color="Maroon" title="c5">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">Q</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="c16">Q</font> )
 <b>by </b><i><a class="ref" href="projred2.html#T19" target="_self" title="PROJRED2:th.19">Th19</a></i>;<br/><a NAME="E4:27_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c8">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">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="c6">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">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="c16">Q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">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="c6">A</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c16">Q</font></span>)</span>
 <b>by </b><i><a class="txt" href="projred2.html#E1:27"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="projred2.html#E3:27"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="projred2.html#E5:27"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="projred2.html#E7:27"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="projred2.html#E9:27"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="projred2.html#E11:27"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="projred2.html#E13:27"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="projred2.html#E15:27"><i><font color="Green" title="E27">A8</font></i></a>, <a class="txt" href="projred2.html#E1:27_1"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="projred2.html#T15" target="_self" title="PROJRED2:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E5:27_1"/>
 ex <font color="Olive" title="b1">Q</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="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">Q</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="b1">Q</font> &amp; not <font color="Maroon" title="c5">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">Q</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c8">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">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="c6">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">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="b1">Q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">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="c6">A</font>,<font color="Maroon" title="c5">q</font>,<font color="Olive" title="b1">Q</font></span>)</span> )
 <b>by </b><i><a class="txt" href="projred2.html#E15:27"><i><font color="Green" title="E27">A8</font></i></a>, <a class="txt" href="projred2.html#E3:27_1"><i><font color="Green" title="E29">A10</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:27"/>
 ex <font color="Olive" title="b1">Q</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="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">Q</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="b1">Q</font> &amp; not <font color="Maroon" title="c5">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">Q</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c8">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">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="c6">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">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="b1">Q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">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="c6">A</font>,<font color="Maroon" title="c5">q</font>,<font color="Olive" title="b1">Q</font></span>)</span> )
 <b>by </b><i><a class="txt" href="projred2.html#E1:27"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>


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