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<b>let </b><font color="Maroon" title="c1">IPP</font> be  <a href="incproj.html#V5" title="INCPROJ:attr.5">2-dimensional</a> <a href="incproj.html#V9" title="INCPROJ:attr.9">Desarguesian</a> <a href="incproj.html#NM2" title="INCPROJ:NM.2">IncProjSp</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">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="c5">A</font>, <font color="Maroon" title="c6">B</font>, <font color="Maroon" title="c7">C</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>;<br/>















<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E22">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="c5">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="c6">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="c7">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">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="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> &amp; <font color="Maroon" title="c4">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="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c4">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="c5">A</font>,<font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c7">C</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  )
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">c</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="E3:29"/><i><font color="Green" title="E23">A2</font></i>: 
( <font color="Maroon" title="c9">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c9">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">C</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14" title="INCPROJ:def.14">INCPROJ:def 14</a></i>;<br/>
<div><i><font color="Green" title="E24">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1"/>
not <font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c8">O</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c10">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="E3:29_1"/><i><font color="Green" title="E25">A4</font></i>: 
( <font color="Maroon" title="c9">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">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="c10">Q</font> &amp; not <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">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="c5">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">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="c10">Q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">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="c5">A</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c10">Q</font></span>)</span> )
 <b>by </b><i><a class="txt" href="projred2.html#E1:29"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="projred2.html#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="projred2.html#E29"><i><font color="Green" title="E19">Lm3</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c11">Q</font> = <font color="Maroon" title="c10">Q</font>;<br/><b>thus </b><a NAME="E4:29_1"/>
<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c11">Q</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a> 
 <b>by </b><i><a class="txt" href="projred2.html#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="projred2.html#E3:29_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></i>;<br/><b>thus </b><a NAME="E5:29_1"/>
( 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="c11">Q</font> &amp; not <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">Q</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">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="c5">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">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="c11">Q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">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="c5">A</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c11">Q</font></span>)</span> )
 <b>by </b><i><a class="txt" href="projred2.html#E3:29_1"><i><font color="Green" title="E25">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_2"/>
<font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c8">O</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c10">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="E3:29_2"/><i><font color="Green" title="E25">A5</font></i>: 
( <font color="Maroon" title="c9">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">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="c10">Q</font> &amp; not <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">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="c5">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">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="c10">Q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">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="c5">A</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c10">Q</font></span>)</span> )
 <b>by </b><i><a class="txt" href="projred2.html#E1:29"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="projred2.html#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="projred2.html#E30"><i><font color="Green" title="E20">Lm4</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c11">Q</font> = <font color="Maroon" title="c10">Q</font>;<br/><b>thus </b><a NAME="E4:29_2"/>
<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c11">Q</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a> 
 <b>by </b><i><a class="txt" href="projred2.html#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="projred2.html#E3:29_2"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></i>;<br/><b>thus </b><a NAME="E5:29_2"/>
( 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="c11">Q</font> &amp; not <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">Q</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">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="c5">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">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="c11">Q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">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="c5">A</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c11">Q</font></span>)</span> )
 <b>by </b><i><a class="txt" href="projred2.html#E3:29_2"><i><font color="Green" title="E25">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:29"/>
 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="c5">A</font>,<font color="Maroon" title="c7">C</font>,<font color="Olive" title="b1">Q</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &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="c4">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="c7">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">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="c5">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">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="c6">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="c5">A</font>,<font color="Maroon" title="c4">q</font>,<font color="Olive" title="b1">Q</font></span>)</span> )
 <b>by </b><i><a class="txt" href="projred2.html#E4:29"><i><font color="Green" title="E24">A3</font></i></a></i>;<br/>


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