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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font>, <font color="Olive" title="b2">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font>  st ( 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">K</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">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b3">P</font> ) ) holds <br/><font color="Olive" title="b1">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">L</font></span><br/><b>let </b><font color="Maroon" title="c2">K</font>, <font color="Maroon" title="c3">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:()"><i><font color="Red">::  thesis: </font></i></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">K</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="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">P</font> ) ) implies <font color="Maroon" title="c2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">L</font> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:39"/><i><font color="Green" title="E36">A1</font></i>: 
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">K</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="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">P</font> )
 <b>and </b><br/><a NAME="E2:39"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E4:39"/><i><font color="Green" title="E38">A3</font></i>: 
<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>
 <b>and </b><br/><a NAME="E5:39"/><i><font color="Green" title="E39">A4</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">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">K</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D8" target="_self" title="INCSP_1:def.8">Def8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">C</font>, <font color="Maroon" title="c7">D</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E7:39"/><i><font color="Green" title="E40">A5</font></i>: 
<font color="Maroon" title="c6">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">D</font>
 <b>and </b><br/><a NAME="E8:39"/><i><font color="Green" title="E41">A6</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c7">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="c3">L</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D8" target="_self" title="INCSP_1:def.8">Def8</a></i>;<br/>
<a NAME="E9:39"/>
( <font color="Maroon" title="c4">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c5">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c6">C</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E2:39"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="incsp_1.html#E5:39"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="incsp_1.html#E8:39"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a></i>;<br/>
<a NAME="E10:39"/><b>then </b>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T12" target="_self" title="INCSP_1:th.12">Th12</a></i>;<br/>
<a NAME="E11:39"/><b>then </b>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#NR6" title="INCSP_1:NR.6">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="incsp_1.html#D6" target="_self" title="INCSP_1:def.6">Def6</a></i>;<br/>
<a NAME="E12:39"/><b>then </b>
<span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c7">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> <a href="incsp_1.html#NR7" title="INCSP_1:NR.7">is_coplanar</a> 
 
<b>by </b><i><a class="ref" href="incsp_1.html#T38" target="_self" title="INCSP_1:th.38">Th38</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">Q</font> being   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E14:39"/><i><font color="Green" title="E42">A7</font></i>: 
<span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c7">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">Q</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D7" target="_self" title="INCSP_1:def.7">Def7</a></i>;<br/>
<a NAME="E15:39"/>
( <font color="Maroon" title="c4">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c8">Q</font> &amp; <font color="Maroon" title="c5">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c8">Q</font> &amp; <font color="Maroon" title="c6">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c8">Q</font> &amp; <font color="Maroon" title="c7">D</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c8">Q</font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E14:39"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="incsp_1.html#T15" target="_self" title="INCSP_1:th.15">Th15</a></i>;<br/>
<a NAME="E16:39"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">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">Q</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c7">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">Q</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T13" target="_self" title="INCSP_1:th.13">Th13</a></i>;<br/>
<a NAME="E17:39"/><b>then </b>
( <font color="Maroon" title="c2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> &amp; <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E4:39"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="incsp_1.html#E5:39"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="incsp_1.html#E7:39"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="incsp_1.html#E8:39"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">Def14</a></i>;<br/>
<b>hence </b><a NAME="E18:39"/>
contradiction
 <b>by </b><i><a class="txt" href="incsp_1.html#E1:39"><i><font color="Green" title="E36">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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