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<div class="add">

<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> 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="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>;<br/>















<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E19">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="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> )
 ;<br/>

<a NAME="E2:25"/>
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="25_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_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> 
 ;<br/>

<a NAME="E2:25_1"/><b>then </b>
 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> )
 
<b>by </b><i><a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E3:25_1"/>
contradiction
 <b>by </b><i><a class="txt" href="projred2.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><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><b> such that </b><br/><a NAME="E4:25"/><i><font color="Green" title="E20">A2</font></i>: 
( <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> )
 <b>and </b><br/><a NAME="E5:25"/><i><font color="Green" title="E21">A3</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="projred2.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="projred2.html#T21" target="_self" title="PROJRED2:th.21">Th21</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">q</font>
;<br/>

<b>thus </b><a NAME="E6:25"/>
( <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> )
 <b>by </b><i><a class="txt" href="projred2.html#E4:25"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/>

<a NAME="E7:25"/><i><font color="Green" title="E22">A4</font></i>: 
(  <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> &amp;  <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> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="projred2.html#E4:25"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="projred2.html#T8" target="_self" title="PROJRED2:th.8">Th8</a></i>;<br/>
<a NAME="E8:25"/><i><font color="Green" title="E23">A5</font></i>: 
(  <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> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="projred2.html#T8" target="_self" title="PROJRED2:th.8">Th8</a></i>;<br/>
<b>thus </b><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>
<b>by </b><i><a class="txt" href="projred2.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="projred2.html#T9" target="_self" title="PROJRED2:th.9">Th9</a></i>
<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>
<b>by </b><i><a class="txt" href="projred2.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="projred2.html#T9" target="_self" title="PROJRED2:th.9">Th9</a></i>
<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> 
<b>by </b><i><a class="txt" href="projred2.html#E5:25"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="projred2.html#E8:25"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="funct_1.html#T66" title="FUNCT_1:th.66">FUNCT_1:66</a></i>
<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>
<b>by </b><i><a class="txt" href="projred2.html#E7:25"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="funct_1.html#T66" title="FUNCT_1:th.66">FUNCT_1:66</a></i>
<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>
<b>by </b><i><a class="txt" href="projred2.html#E4:25"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="projred2.html#T9" target="_self" title="PROJRED2:th.9">Th9</a></i>
<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>
<b>by </b><i><a class="txt" href="projred2.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="projred2.html#T9" target="_self" title="PROJRED2:th.9">Th9</a></i>
;<br/>


</div>
