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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="incsp_1.html#NM4" title="INCSP_1:NM.4">IncSpace</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">L</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font>  ex <font color="Olive" title="b2">K</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font> st <br/> for <font color="Olive" title="b3">P</font> being   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font>  holds <br/>(  not <font color="Olive" title="b1">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b3">P</font> or  not <font color="Olive" title="b2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b3">P</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">L</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">K</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font> st <br/> for <font color="Olive" title="b2">P</font> being   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font>  holds <br/>(  not <font color="Maroon" title="c2">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b2">P</font> or  not <font color="Olive" title="b1">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b2">P</font> )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E2:75"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">and </span><br/><a NAME="E3:75"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></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="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#D8" target="_self" title="INCSP_1:def.8">Def8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">C</font>, <font color="Maroon" title="c6">D</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E5:75"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
 not <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c6">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> is  <a href="incsp_1.html#V4" title="INCSP_1:attr.4">planar</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="incsp_1.html#T47" target="_self" title="INCSP_1:th.47">Th77</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c7">K</font> =  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c6">D</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">P</font> being   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font>  holds <br/>(  not <font color="Maroon" title="c2">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">P</font> or  not <font color="Maroon" title="c7">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">P</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c8">P</font> being   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><a NAME="E7:75"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
<font color="Maroon" title="c2">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">P</font>
 <span class="kw">and </span><br/><a NAME="E8:75"/><span class="lab"><font color="Green" title="E59">A5</font></span>: 
<font color="Maroon" title="c7">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">P</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="E9:75"/>
 not <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> is  <a href="incsp_1.html#V4" title="INCSP_1:attr.4">planar</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="enumset1.html#T73" title="ENUMSET1:th.73">ENUMSET1:73</a></span>;<br/>
<a NAME="E10:75"/><span class="kw">then </span>
<font color="Maroon" title="c5">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T16" target="_self" title="INCSP_1:th.16">Th37</a></span>;<br/>
<a NAME="E11:75"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c6">D</font></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="c7">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#D19" target="_self" title="INCSP_1:def.19">Def19</a></span>;<br/>
<a NAME="E12:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">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="c5">C</font>,<font color="Maroon" title="c6">D</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c8">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:75"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th35</a></span>;<br/>
<a NAME="E13:75"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c8">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E7:75"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th35</a></span>;<br/>
<a NAME="E14:75"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c6">D</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c8">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:75"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th21</a></span>;<br/>
<a NAME="E15:75"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c6">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c8">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T5" title="ENUMSET1:th.5">ENUMSET1:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:75"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="incsp_1.html#D7" target="_self" title="INCSP_1:def.7">Def7</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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