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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="projpl_1.html#NM1" title="PROJPL_1:NM.1">IncProjectivePlane</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">q1</font>, <font color="Olive" title="b2">q2</font>, <font color="Olive" title="b3">r1</font>, <font color="Olive" title="b4">r2</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b5">H</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b3">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">r2</font> &amp; <font color="Olive" title="b3">r1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b5">H</font> &amp; <font color="Olive" title="b4">r2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b5">H</font> &amp; <font color="Olive" title="b1">q1</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Olive" title="b5">H</font> &amp; <font color="Olive" title="b2">q2</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Olive" title="b5">H</font> holds <br/><font color="Olive" title="b1">q1</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Olive" title="b3">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">q2</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Olive" title="b4">r2</font></span><br/><b>let </b><font color="Maroon" title="c2">q1</font>, <font color="Maroon" title="c3">q2</font>, <font color="Maroon" title="c4">r1</font>, <font color="Maroon" title="c5">r2</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">H</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">G</font>  st <font color="Maroon" title="c4">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">r2</font> &amp; <font color="Maroon" title="c4">r1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">H</font> &amp; <font color="Maroon" title="c5">r2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">H</font> &amp; <font color="Maroon" title="c2">q1</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Olive" title="b1">H</font> &amp; <font color="Maroon" title="c3">q2</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Olive" title="b1">H</font> holds <br/><font color="Maroon" title="c2">q1</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c4">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q2</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c5">r2</font></span><br/><b>let </b><font color="Maroon" title="c6">H</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">r2</font> &amp; <font color="Maroon" title="c4">r1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">H</font> &amp; <font color="Maroon" title="c5">r2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">H</font> &amp; <font color="Maroon" title="c2">q1</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Maroon" title="c6">H</font> &amp; <font color="Maroon" title="c3">q2</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Maroon" title="c6">H</font> implies <font color="Maroon" title="c2">q1</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c4">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q2</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c5">r2</font> )</span><br/>





<b>assume </b><a NAME="E1:44"/><i><font color="Green" title="E33">A1</font></i>: 
( <font color="Maroon" title="c4">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">r2</font> &amp; <font color="Maroon" title="c4">r1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">H</font> &amp; <font color="Maroon" title="c5">r2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">H</font> &amp; <font color="Maroon" title="c2">q1</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Maroon" title="c6">H</font> &amp; <font color="Maroon" title="c3">q2</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Maroon" title="c6">H</font> &amp; <font color="Maroon" title="c2">q1</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c4">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q2</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c5">r2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>set </b><font color="Maroon" title="c7">L1</font> = <font color="Maroon" title="c2">q1</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c4">r1</font>;<br/>
<b>set </b><font color="Maroon" title="c8">L2</font> = <font color="Maroon" title="c3">q2</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c5">r2</font>;<br/>
<a NAME="E2:44"/><i><font color="Green" title="E34">A2</font></i>: 
( <font color="Maroon" title="c2">q1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">q1</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c4">r1</font> &amp; <font color="Maroon" title="c3">q2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">q2</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c5">r2</font> &amp; <font color="Maroon" title="c4">r1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">q1</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c4">r1</font> &amp; <font color="Maroon" title="c5">r2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">q2</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c5">r2</font> )
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="projpl_1.html#T16" target="_self" title="PROJPL_1:th.16">Th16</a></i>;<br/>
<a NAME="E3:44"/><b>then </b>
( <font color="Maroon" title="c4">r1</font> <a href="projpl_1.html#R2" title="PROJPL_1:pred.2">on</a> <font color="Maroon" title="c2">q1</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c4">r1</font>,<font color="Maroon" title="c6">H</font> &amp; <font color="Maroon" title="c5">r2</font> <a href="projpl_1.html#R2" title="PROJPL_1:pred.2">on</a> <font color="Maroon" title="c3">q2</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c5">r2</font>,<font color="Maroon" title="c6">H</font> )
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="projpl_1.html#D2" target="_self" title="PROJPL_1:def.2">Def2</a></i>;<br/>
<a NAME="E4:44"/><b>then </b>
( <font color="Maroon" title="c4">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q1</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c4">r1</font></span>)</span> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c6">H</font> &amp; <font color="Maroon" title="c5">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q2</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c5">r2</font></span>)</span> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c6">H</font> )
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="#E2:44"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="projpl_1.html#D9" target="_self" title="PROJPL_1:def.9">Def9</a></i>;<br/>
<b>hence </b><a NAME="E5:44"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E33">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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