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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">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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:41"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:41"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:41"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:41"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">set </span><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/>
<span class="kw">set </span><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><span class="lab"><font color="Green" title="E37">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:41_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E36">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="E2:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A6</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/><a NAME="E3:41_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E36">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="E4:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A7</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="projpl_1.html#D8" target="_self" title="PROJPL_1:def.8">Def8</a></span>;<br/><span class="kw">assume </span><a NAME="E5:41_1"/><span class="lab"><font color="Green" title="E40">A8</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E6:41_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E36">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="E7:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A9</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/><a NAME="E8:41_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="projpl_1.html#T24" target="_self" title="PROJPL_1:th.24">Th24</a></span>;<br/><span class="kw">hence </span><a NAME="E9:41_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E5:41_1"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E7:41_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E4:41_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="projpl_1.html#D8" target="_self" title="PROJPL_1:def.8">Def8</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/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:41_2"/><span class="lab"><font color="Green" title="E38">A10</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="projpl_1.html#T24" target="_self" title="PROJPL_1:th.24">Th24</a></span>;<br/><span class="kw">hence </span><a NAME="E3:41_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E1:41_2"><span class="lab"><font color="Green" title="E38">A10</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:41"/>
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E37">A5</font></span></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/>


</div>
