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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">e</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="b2">I</font>, <font color="Olive" title="b3">L1</font>, <font color="Olive" title="b4">L2</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="b1">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">L1</font> &amp; <font color="Olive" title="b1">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">L2</font> &amp; <font color="Olive" title="b3">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">L2</font> &amp; <font color="Olive" title="b1">e</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Olive" title="b2">I</font> holds <br/>( <font color="Olive" title="b2">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Olive" title="b3">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Olive" title="b4">L2</font> &amp; <font color="Olive" title="b3">L1</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Olive" title="b2">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">L2</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Olive" title="b2">I</font> )</span><br/><b>let </b><font color="Maroon" title="c2">e</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">I</font>, <font color="Olive" title="b2">L1</font>, <font color="Olive" title="b3">L2</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="c2">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">L1</font> &amp; <font color="Maroon" title="c2">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">L2</font> &amp; <font color="Olive" title="b2">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">L2</font> &amp; <font color="Maroon" title="c2">e</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Olive" title="b1">I</font> holds <br/>( <font color="Olive" title="b1">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Olive" title="b2">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Olive" title="b3">L2</font> &amp; <font color="Olive" title="b2">L1</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Olive" title="b1">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">L2</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Olive" title="b1">I</font> )</span><br/><b>let </b><font color="Maroon" title="c3">I</font>, <font color="Maroon" title="c4">L1</font>, <font color="Maroon" title="c5">L2</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="c2">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">L1</font> &amp; <font color="Maroon" title="c2">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">L2</font> &amp; <font color="Maroon" title="c4">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">L2</font> &amp; <font color="Maroon" title="c2">e</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Maroon" title="c3">I</font> implies ( <font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c4">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">L2</font> &amp; <font color="Maroon" title="c4">L1</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">L2</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c3">I</font> ) )</span><br/>





<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E33">A1</font></i>: 
( <font color="Maroon" title="c2">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">L1</font> &amp; <font color="Maroon" title="c2">e</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">L2</font> &amp; <font color="Maroon" title="c4">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">L2</font> &amp; <font color="Maroon" title="c2">e</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Maroon" title="c3">I</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c4">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">L2</font> &amp; <font color="Maroon" title="c4">L1</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">L2</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c3">I</font> )</span><br/>

<b>set </b><font color="Maroon" title="c6">p1</font> = <font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c4">L1</font>;<br/>
<b>set </b><font color="Maroon" title="c7">p2</font> = <font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">L2</font>;<br/>
<div><i><font color="Green" title="E34">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1"/><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c4">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">L2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:41_1"/><i><font color="Green" title="E36">A4</font></i>: 
( <font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c4">L1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">L2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c4">L1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">L1</font> &amp; <font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">L2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">L2</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:41"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="projpl_1.html#T24" target="_self" title="PROJPL_1:th.24">Th24</a></i>;<br/><a NAME="E3:41_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">e</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c4">L1</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c4">L1</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">e</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">L2</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c5">L2</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:41"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></i>;<br/><a NAME="E4:41_1"/><b>then </b>
( <font color="Maroon" title="c4">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c4">L1</font></span>)</span> &amp; <font color="Maroon" title="c5">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">L2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:41"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="projpl_1.html#E2:41_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="projpl_1.html#D8" target="_self" title="PROJPL_1:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E5:41_1"/>
contradiction
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:41"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="projpl_1.html#E1:41_1"><i><font color="Green" title="E35">A3</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/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_2"/><i><font color="Green" title="E35">A5</font></i>: 
<font color="Maroon" title="c4">L1</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L2</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c3">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:41_2"/>
<font color="Maroon" title="c4">L1</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c4">L1</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:41"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="projpl_1.html#T24" target="_self" title="PROJPL_1:th.24">Th24</a></i>;<br/><b>hence </b><a NAME="E3:41_2"/>
contradiction
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:41"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="projpl_1.html#E2:41"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="projpl_1.html#E1:41_2"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="projpl_1.html#T24" target="_self" title="PROJPL_1:th.24">Th24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:41"/>
( <font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c4">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">L2</font> &amp; <font color="Maroon" title="c4">L1</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">L2</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c3">I</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:41"><i><font color="Green" title="E34">A2</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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