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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">q</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="b3">A</font>, <font color="Olive" title="b4">B</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">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">B</font> &amp; <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">A</font> &amp; <font color="Olive" title="b2">q</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Olive" title="b3">A</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">A</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Olive" title="b4">B</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Olive" title="b4">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">B</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Olive" title="b4">B</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Olive" title="b3">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">q</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">B</font> &amp; <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c3">q</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">A</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Olive" title="b2">B</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Olive" title="b2">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">B</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Olive" title="b2">B</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Olive" title="b1">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c3">q</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">A</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">B</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">B</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">B</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Maroon" title="c4">A</font> ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">and </span><br/><a NAME="E2:35"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E3:35"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E4:35"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">A</font> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">B</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">B</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Maroon" title="c4">A</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">D</font> = <font color="Maroon" title="c3">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">a</font>;<br/>
<a NAME="E5:35"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
( <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="projpl_1.html#T16" target="_self" title="PROJPL_1:th.16">Th16</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">d</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">B</font>;<br/>
<a NAME="E6:35"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="projpl_1.html#V1" title="PROJPL_1:attr.1">configuration</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="projpl_1.html#T23" target="_self" title="PROJPL_1:th.23">Th23</a></span>;<br/>
<a NAME="E7:35"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Maroon" title="c5">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="projpl_1.html#T24" target="_self" title="PROJPL_1:th.24">Th24</a></span>;<br/>
<a NAME="E8:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A8</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="projpl_1.html#T16" target="_self" title="PROJPL_1:th.16">Th16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:35"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projpl_1.html#T24" target="_self" title="PROJPL_1:th.24">Th24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">B</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Maroon" title="c4">A</font></span><br/>

<span class="kw">assume </span><a NAME="E10:35"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E11:35"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="ref" href="projpl_1.html#T24" target="_self" title="PROJPL_1:th.24">Th24</a></span>;<br/>
<a NAME="E12:35"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="projpl_1.html#K1" title="PROJPL_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="projpl_1.html#K2" title="PROJPL_1:func.2">*</a> <font color="Maroon" title="c5">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="txt" href="#E10:35"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="projpl_1.html#D4" target="_self" title="PROJPL_1:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:35"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E8:35"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="ref" href="projpl_1.html#T24" target="_self" title="PROJPL_1:th.24">Th24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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