<?xml version="1.0"?>
<div class="add">

<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>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">m'</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="b4">I</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="b2">m</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">I</font> &amp; <font color="Olive" title="b3">m'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">I</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">m'</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="b4">I</font> holds <br/>( <font color="Olive" title="b2">m</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">m'</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Olive" title="b1">e</font> &amp; <font color="Olive" title="b1">e</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">e</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Olive" title="b3">m'</font> )</span><br/><b>let </b><font color="Maroon" title="c2">e</font>, <font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">m'</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> 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="c3">m</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">I</font> &amp; <font color="Maroon" title="c4">m'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">I</font> &amp; <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">m'</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="Maroon" title="c3">m</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">m'</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</font> &amp; <font color="Maroon" title="c2">e</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">e</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c4">m'</font> )</span><br/><b>let </b><font color="Maroon" title="c5">I</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="c3">m</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">I</font> &amp; <font color="Maroon" title="c4">m'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">I</font> &amp; <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">m'</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="c5">I</font> implies ( <font color="Maroon" title="c3">m</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">m'</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</font> &amp; <font color="Maroon" title="c2">e</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">e</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c4">m'</font> ) )</span><br/>





<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E33">A1</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">I</font> &amp; <font color="Maroon" title="c4">m'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">I</font> &amp; <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">m'</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="c5">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">m</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">m'</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</font> &amp; <font color="Maroon" title="c2">e</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">e</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c4">m'</font> )</span><br/>

<b>set </b><font color="Maroon" title="c6">L1</font> = <font color="Maroon" title="c3">m</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</font>;<br/>
<b>set </b><font color="Maroon" title="c7">L2</font> = <font color="Maroon" title="c4">m'</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</font>;<br/>
<div><i><font color="Green" title="E34">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1"/><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m'</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</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="E2:40_1"/><i><font color="Green" title="E36">A4</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">m</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</font> &amp; <font color="Maroon" title="c4">m'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">m'</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</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="c3">m</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</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="c4">m'</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:40"><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="E3:40_1"/><b>then </b>
( <font color="Maroon" title="c3">m</font> <a href="projpl_1.html#R2" title="PROJPL_1:pred.2">on</a> <font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c3">m</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</font> &amp; <font color="Maroon" title="c4">m'</font> <a href="projpl_1.html#R2" title="PROJPL_1:pred.2">on</a> <font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c4">m'</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:40"><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="E4:40_1"/><b>then </b>
( <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</font></span>)</span> &amp; <font color="Maroon" title="c4">m'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">I</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m'</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</font></span>)</span> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:40"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="projpl_1.html#E2:40_1"><i><font color="Green" title="E36">A4</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="E5:40_1"/>
contradiction
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:40"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="projpl_1.html#E1:40_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="40_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_2"/><i><font color="Green" title="E35">A5</font></i>: 
<font color="Maroon" title="c2">e</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c3">m</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> <font color="Maroon" title="c4">m'</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="E2:40_2"/>
<font color="Maroon" title="c3">m</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</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> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:40"><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/><b>hence </b><a NAME="E3:40_2"/>
contradiction
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:40"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="projpl_1.html#E2:40"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="projpl_1.html#E1:40_2"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="projpl_1.html#T16" target="_self" title="PROJPL_1:th.16">Th16</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="E4:40"/>
( <font color="Maroon" title="c3">m</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">m'</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">e</font> &amp; <font color="Maroon" title="c2">e</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">e</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c4">m'</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:40"><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/>


</div>
