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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">o1</font>, <font color="Olive" title="b2">o2</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">O1</font>, <font color="Olive" title="b4">O2</font>, <font color="Olive" title="b5">O3</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">o1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">O1</font> &amp; not <font color="Olive" title="b1">o1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">O2</font> &amp; not <font color="Olive" title="b2">o2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">O2</font> &amp; not <font color="Olive" title="b2">o2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b5">O3</font> &amp; <font color="Olive" title="b3">O1</font>,<font color="Olive" title="b4">O2</font>,<font color="Olive" title="b5">O3</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; <font color="Olive" title="b3">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">O3</font> holds <br/> ex <font color="Olive" title="b6">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="b6">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">O1</font> &amp; not <font color="Olive" title="b6">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b5">O3</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Olive" title="b4">O2</font>,<font color="Olive" title="b2">o2</font>,<font color="Olive" title="b5">O3</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">O1</font>,<font color="Olive" title="b1">o1</font>,<font color="Olive" title="b4">O2</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="b3">O1</font>,<font color="Olive" title="b6">o</font>,<font color="Olive" title="b5">O3</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">o1</font>, <font color="Maroon" title="c3">o2</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">O1</font>, <font color="Olive" title="b2">O2</font>, <font color="Olive" title="b3">O3</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">o1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">O1</font> &amp; not <font color="Maroon" title="c2">o1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">O2</font> &amp; not <font color="Maroon" title="c3">o2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">O2</font> &amp; not <font color="Maroon" title="c3">o2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">O3</font> &amp; <font color="Olive" title="b1">O1</font>,<font color="Olive" title="b2">O2</font>,<font color="Olive" title="b3">O3</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; <font color="Olive" title="b1">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">O3</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">O1</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">O3</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Olive" title="b2">O2</font>,<font color="Maroon" title="c3">o2</font>,<font color="Olive" title="b3">O3</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">O1</font>,<font color="Maroon" title="c2">o1</font>,<font color="Olive" title="b2">O2</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">O1</font>,<font color="Olive" title="b4">o</font>,<font color="Olive" title="b3">O3</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">O1</font>, <font color="Maroon" title="c5">O2</font>, <font color="Maroon" title="c6">O3</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">o1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">O1</font> &amp; not <font color="Maroon" title="c2">o1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">O2</font> &amp; not <font color="Maroon" title="c3">o2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">O2</font> &amp; not <font color="Maroon" title="c3">o2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">O3</font> &amp; <font color="Maroon" title="c4">O1</font>,<font color="Maroon" title="c5">O2</font>,<font color="Maroon" title="c6">O3</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; <font color="Maroon" title="c4">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">O3</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="c4">O1</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="c6">O3</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">O2</font>,<font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c6">O3</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">O1</font>,<font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c5">O2</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="c4">O1</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c6">O3</font> ) )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
( not <font color="Maroon" title="c2">o1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">O1</font> &amp; not <font color="Maroon" title="c2">o1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">O2</font> &amp; not <font color="Maroon" title="c3">o2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">O2</font> &amp; not <font color="Maroon" title="c3">o2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">O3</font> )
 <span class="kw">and </span><br/><a NAME="E2:16"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon" title="c4">O1</font>,<font color="Maroon" title="c5">O2</font>,<font color="Maroon" title="c6">O3</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a> 
 <span class="kw">and </span><br/><a NAME="E3:16"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c4">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">O3</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="c4">O1</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="c6">O3</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">O2</font>,<font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c6">O3</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">O1</font>,<font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c5">O2</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="c4">O1</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c6">O3</font> )</span><br/>

<a NAME="E4:16"/>
 ex <font color="Olive" title="b1">p</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">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">O1</font> &amp; <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">O2</font> &amp; <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">O3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16"/>
 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="c4">O1</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="c6">O3</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c5">O2</font>,<font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c6">O3</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">O1</font>,<font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c5">O2</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="c4">O1</font>,<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c6">O3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="projred1.html#T28" title="PROJRED1:th.28">PROJRED1:28</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/>


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