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<b>let </b><font color="Maroon">G</font> be   <a href="projpl_1.html#NM1">IncProjectivePlane</a>;<br/><b>let </b><font color="Maroon">q1</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>, <font color="Maroon">q2</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>, <font color="Maroon">r1</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>, <font color="Maroon">r2</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">H</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>;<br/>





<b>assume </b><a NAME="E1:44"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">r1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">H</font> &amp; <font color="Maroon">r2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">H</font> &amp; <font color="Maroon">q1</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">H</font> &amp; <font color="Maroon">q2</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">H</font> &amp; <font color="Maroon">q1</font> <a href="projpl_1.html#K1" target="_self">*</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="projpl_1.html#K1" target="_self">*</a> <font color="Maroon">r2</font> )
 ;<br/>

<b>set </b><font color="Maroon">L1</font> = <font color="Maroon">q1</font> <a href="projpl_1.html#K1" target="_self">*</a> <font color="Maroon">r1</font>;<br/>
<b>set </b><font color="Maroon">L2</font> = <font color="Maroon">q2</font> <a href="projpl_1.html#K1" target="_self">*</a> <font color="Maroon">r2</font>;<br/>
<a NAME="E2:44"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">q1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">q1</font> <a href="projpl_1.html#K1" target="_self">*</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">q2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">q2</font> <a href="projpl_1.html#K1" target="_self">*</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">r1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">q1</font> <a href="projpl_1.html#K1" target="_self">*</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">q2</font> <a href="projpl_1.html#K1" target="_self">*</a> <font color="Maroon">r2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:44"/><b>then </b>
( <font color="Maroon">r1</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">q1</font> <a href="projpl_1.html#K1" target="_self">*</a> <font color="Maroon">r1</font>,<font color="Maroon">H</font> &amp; <font color="Maroon">r2</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">q2</font> <a href="projpl_1.html#K1" target="_self">*</a> <font color="Maroon">r2</font>,<font color="Maroon">H</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:44"/><b>then </b>
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="projpl_1.html#K1" target="_self">*</a> <font color="Maroon">r1</font></span>)</span> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">H</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="projpl_1.html#K1" target="_self">*</a> <font color="Maroon">r2</font></span>)</span> <a href="projpl_1.html#K2" target="_self">*</a> <font color="Maroon">H</font> )
 
<b>by </b><i>, <a class="ref" href="projpl_1.html#D4" target="_self">Def4</a>, <a class="ref" href="projpl_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:44"/>
contradiction
 <b>by </b><i/>;<br/>


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