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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> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</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> holds <br/> ex <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> st 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> 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"> ( 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> implies  ex <font color="Olive" title="b1">D</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st 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="Olive" title="b1">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><a NAME="E1:67"/><i><font color="Green" title="E53">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not for <font color="Olive" title="b1">D</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> holds  <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="Olive" title="b1">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>consider </b><font color="Maroon" title="c5">P</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="E3:67"/><i><font color="Green" title="E54">A2</font></i>: 
<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="c5">P</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D12" target="_self" title="INCSP_1:def.12">Def12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">A1</font>, <font color="Maroon" title="c7">B1</font>, <font color="Maroon" title="c8">C1</font>, <font color="Maroon" title="c9">D1</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="E5:67"/><i><font color="Green" title="E55">A3</font></i>: 
not <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c6">A1</font>,<font color="Maroon" title="c7">B1</font>,<font color="Maroon" title="c8">C1</font>,<font color="Maroon" title="c9">D1</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>
 <b>by </b><i><a class="ref" href="incsp_1.html#D16" target="_self" title="INCSP_1:def.16">Def16</a></i>;<br/>
<a NAME="E6:67"/>
not <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c6">A1</font>,<font color="Maroon" title="c7">B1</font>,<font color="Maroon" title="c8">C1</font>,<font color="Maroon" title="c9">D1</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="c5">P</font>
 
<b>by </b><i><a class="txt" href="#E5:67"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="incsp_1.html#D7" target="_self" title="INCSP_1:def.7">Def7</a></i>;<br/>
<a NAME="E7:67"/><b>then </b>
(  not <font color="Maroon" title="c6">A1</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c5">P</font> or  not <font color="Maroon" title="c7">B1</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c5">P</font> or  not <font color="Maroon" title="c8">C1</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c5">P</font> or  not <font color="Maroon" title="c9">D1</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c5">P</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T15" target="_self" title="INCSP_1:th.15">Th15</a></i>;<br/>
<a NAME="E8:67"/><b>then </b>
(  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="c6">A1</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> or  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="c7">B1</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> or  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="c8">C1</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> or  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="c9">D1</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> )
 
<b>by </b><i><a class="txt" href="#E1:67"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="#E3:67"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="incsp_1.html#T40" target="_self" title="INCSP_1:th.40">Th40</a></i>;<br/>
<b>hence </b><a NAME="E9:67"/>
not for <font color="Olive" title="b1">D</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> holds  <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="Olive" title="b1">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>
 ; <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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