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

<b>let </b><font color="Maroon">G</font> be   <a href="projpl_1.html#NM1">IncProjectivePlane</a>;<br/><b>let </b><font color="Maroon">e</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">I</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>, <font color="Maroon">L1</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>, <font color="Maroon">L2</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>;<br/>





<b>assume </b><a NAME="E1:41"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">e</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">e</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">L1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">e</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">I</font> )
 ;<br/>

<b>set </b><font color="Maroon">p1</font> = <font color="Maroon">I</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">L1</font>;<br/>
<b>set </b><font color="Maroon">p2</font> = <font color="Maroon">I</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">L2</font>;<br/>
<div><i><font color="Green">E37</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">E40</font></i>: 
<font color="Maroon">I</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">L2</font>
 ;<br/><a NAME="E2:41_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">I</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">L1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">I</font> &amp; <font color="Maroon">I</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">L2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">I</font> &amp; <font color="Maroon">I</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">L1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">I</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">L2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">L2</font> )
 <b>by </b><i>, <a class="ref" href="projpl_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E3:41_1"/><b>then </b>
( <font color="Maroon">e</font>,<font color="Maroon">I</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">L1</font> <a href="incproj.html#R3">on</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">e</font>,<font color="Maroon">I</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">L2</font> <a href="incproj.html#R3">on</a> <font color="Maroon">L2</font> )
 <b>by </b><i>, <a class="ref" href="incproj.html#D7">INCPROJ:def 7</a></i>;<br/><a NAME="E4:41_1"/><b>then </b>
( <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> <a href="projpl_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">L1</font></span>)</span> &amp; <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> <a href="projpl_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">L2</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="projpl_1.html#E1:41"><i><font color="Green">E36</font></i></a>, </i>;<br/><b>hence </b><a NAME="E5:41_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="projpl_1.html#T2" target="_self">Th2</a></i>;<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">E42</font></i>: 
<font color="Maroon">L1</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:41_2"/>
<font color="Maroon">L1</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">L1</font>
 <b>by </b><i>, <a class="ref" href="projpl_1.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E3:41_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="projpl_1.html#D4" target="_self">Def4</a>, <a class="ref" href="projpl_1.html#T2" target="_self">Th2</a>, <a class="ref" href="projpl_1.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:41"/>
( <font color="Maroon">I</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">L1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">I</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">L1</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">I</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">L2</font> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">I</font> )
 <b>by </b><i><a class="ref" href="projpl_1.html#D4" target="_self">Def4</a></i>;<br/>


</div>
