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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">A</font>, <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">C</font>, <font color="Olive" title="b4">D</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b5">P</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font>  st not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> is <a href="incsp_1.html#V3" title="INCSP_1:attr.3">linear</a> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Olive" title="b5">P</font> &amp; not <font color="Olive" title="b4">D</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b5">P</font> holds <br/> not <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">C</font>,<font color="Olive" title="b4">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><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">C</font>, <font color="Maroon" title="c5">D</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</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>  st not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> is <a href="incsp_1.html#V3" title="INCSP_1:attr.3">linear</a> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Olive" title="b1">P</font> &amp; not <font color="Maroon" title="c5">D</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b1">P</font> holds <br/> not <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">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><br/><b>let </b><font color="Maroon" title="c6">P</font> be   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</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"> ( not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> is <a href="incsp_1.html#V3" title="INCSP_1:attr.3">linear</a> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c6">P</font> &amp; not <font color="Maroon" title="c5">D</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> implies  not <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">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><br/>











<b>assume </b><b>that </b><br/><a NAME="E1:36"/><i><font color="Green" title="E36">A1</font></i>: 
( not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> is <a href="incsp_1.html#V3" title="INCSP_1:attr.3">linear</a> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c6">P</font> )
 <b>and </b><br/><a NAME="E2:36"/><i><font color="Green" title="E37">A2</font></i>: 
not <font color="Maroon" title="c5">D</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">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><br/>

<b>given </b><font color="Maroon" title="c7">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><a NAME="E4:36"/><i><font color="Green" title="E38">A3</font></i>: 
<span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">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="c7">Q</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="incsp_1.html#D7" target="_self" title="INCSP_1:def.7">INCSP_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:36"/>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">D</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c7">Q</font>
 
<b>by </b><i><a class="txt" href="#E4:36"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="enumset1.html#T46" title="ENUMSET1:th.46">ENUMSET1:46</a></i>;<br/>
<a NAME="E6:36"/><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="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c7">Q</font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T19" target="_self" title="INCSP_1:th.19">Th19</a></i>;<br/>
<a NAME="E7:36"/><b>then </b>
<font color="Maroon" title="c6">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Q</font>
 
<b>by </b><i><a class="txt" href="#E1:36"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="incsp_1.html#D13" target="_self" title="INCSP_1:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E8:36"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:36"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="#E4:36"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="incsp_1.html#T15" target="_self" title="INCSP_1:th.15">Th15</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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