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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">c</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">q</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">O</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/>

















<b>assume </b><a NAME="E1:27"/><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; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</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> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; not <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</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; 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>  )
 ;<br/>

<b>consider </b><font color="Maroon">d</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E3:27"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14">INCPROJ:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon">O1</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E5:27"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O1</font> &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O1</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">O3</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E7:27"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O3</font> &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O3</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E9:27"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O1</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14">INCPROJ:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon">O2</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E11:27"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O2</font> &amp; <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O2</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">pp'</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E13:27"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">pp'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O3</font> &amp; <font color="Maroon">pp'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O2</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14">INCPROJ:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon">Q</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E15:27"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">pp'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">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">E67</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E2:27_1"/><b>then </b>
( not <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</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">B</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">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</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; 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>  &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">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">p</font> <a href="incproj.html#R3">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="incproj.html#R3">on</a> <font color="Maroon">C</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">q</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">pp'</font> <a href="incproj.html#R3">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">d</font>,<font color="Maroon">p</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O1</font> &amp; <font color="Maroon">q</font>,<font color="Maroon">p</font>,<font color="Maroon">pp'</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O2</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">d</font>,<font color="Maroon">pp'</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O3</font> )
 <b>by </b><i>, , , , , , , , <a class="ref" href="incproj.html#D7">INCPROJ:def 7</a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/><a NAME="E3:27_1"/><b>then </b><i><font color="Green">E68</font></i>: 
( <font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">A</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C</font> &amp; not <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; not <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E4:27_1"/><b>then </b>
<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="Maroon">q</font>,<font color="Maroon">Q</font></span>)</span>
 <b>by </b><i>, , , , , , , , , </i>;<br/><b>hence </b><a NAME="E5:27_1"/>
 ex <font color="Olive">Q</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">Q</font> &amp; not <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">Q</font> &amp; not <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">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="Olive">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="Maroon">q</font>,<font color="Olive">Q</font></span>)</span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:27"/>
 ex <font color="Olive">Q</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">Q</font> &amp; not <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">Q</font> &amp; not <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">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="Olive">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="Maroon">q</font>,<font color="Olive">Q</font></span>)</span> )
 <b>by </b><i/>;<br/>


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