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<b>let </b><font color="Maroon">IPP</font> be  <a href="incproj.html#V6">2-dimensional</a> <a href="incproj.html#V10">Desarguesian</a> <a href="incproj.html#NM2">IncProjSp</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">Q</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">O</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/>















<b>assume </b><a NAME="E1:24"/><i><font color="Green">E47</font></i>: 
( not <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; not <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; not <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font> <a href="projred2.html#R1" target="_self">are_concurrent</a>  &amp; <font color="Maroon">A</font>,<font color="Maroon">C</font>,<font color="Maroon">Q</font> <a href="projred2.html#R1" target="_self">are_concurrent</a>  &amp; not <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> )
 ;<br/>

<a NAME="E2:24"/><b>then </b>
( not <font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">O</font> <a href="projred2.html#R1" target="_self">are_concurrent</a>  implies  ex <font color="Olive">q</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; not <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; not <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">C</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">a</font>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Olive">q</font>,<font color="Maroon">Q</font></span>)</span> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:24"/>
 ex <font color="Olive">q</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; not <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; not <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">C</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">a</font>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">Q</font>,<font color="Maroon">b</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Olive">q</font>,<font color="Maroon">Q</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>


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