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<div class="add">

<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">P</font> being   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font>  ex <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font>, <font color="Olive" title="b4">C</font>, <font color="Olive" title="b5">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 <br/>( <font color="Olive" title="b2">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b1">P</font> &amp;  not <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font>,<font color="Olive" title="b4">C</font>,<font color="Olive" title="b5">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/><span class="kw">let </span><font color="Maroon" title="c2">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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> st <br/>( <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp;  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/>



<span class="kw">consider </span><font color="Maroon" title="c3">A</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="E2:63"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#D11" target="_self" title="INCSP_1:def.11">Def11</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">A1</font>, <font color="Maroon" title="c5">B1</font>, <font color="Maroon" title="c6">C1</font>, <font color="Maroon" title="c7">D1</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="E4:63"/><span class="lab"><font color="Green" title="E50">A2</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="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</font>,<font color="Maroon" title="c7">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#D16" target="_self" title="INCSP_1:def.16">Def16</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:63_1"/><span class="lab"><font color="Green" title="E51">A3</font></span>: 
 not <font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</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">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> st <br/>( <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp;  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/><div><span class="lab"><font color="Green" title="E52">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:63_1_1"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
<font color="Maroon" title="c4">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="incsp_1.html#T16" target="_self" title="INCSP_1:th.16">Th37</a></span>;<br/><a NAME="E2:63_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">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="c4">A1</font>,<font color="Maroon" title="c5">B1</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>  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</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="E3:63_1_1"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</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>  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="ref" href="incsp_1.html#D19" target="_self" title="INCSP_1:def.19">Def19</a></span>;<br/><a NAME="E4:63_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">C1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a>  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>) implies <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</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#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">C1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a>  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T8" target="_self" title="INCSP_1:th.8">Th18</a></span>;<br/><a NAME="E5:63_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A7</font></span>: 
( <font color="Maroon" title="c6">C1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a>  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>) implies <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</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>  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T3" title="ENUMSET1:th.3">ENUMSET1:3</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c8">Q</font> =  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</font>);<br/><span class="kw">assume </span><a NAME="E6:63_1_1"/><span class="lab"><font color="Green" title="E56">A8</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a>  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</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">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> st <br/>( <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp;  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/><a NAME="E7:63_1_1"/><span class="lab"><font color="Green" title="E57">A9</font></span>: 
 not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="incsp_1.html#T17" target="_self" title="INCSP_1:th.17">Th38</a></span>;<br/><a NAME="E8:63_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</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>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#D20" target="_self" title="INCSP_1:def.20">Def20</a></span>;<br/><a NAME="E9:63_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A10</font></span>: 
( <font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</font>) &amp; <font color="Maroon" title="c6">C1</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T4" target="_self" title="INCSP_1:th.4">Th14</a></span>;<br/><a NAME="E10:63_1_1"/><span class="lab"><font color="Green" title="E59">A11</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</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>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:63_1_1"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="ref" href="incsp_1.html#D20" target="_self" title="INCSP_1:def.20">Def20</a></span>;<br/><a NAME="E11:63_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">D1</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</font>) implies <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</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="c7">D1</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>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T9" target="_self" title="INCSP_1:th.9">Th19</a></span>;<br/><a NAME="E12:63_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A12</font></span>: 
( <font color="Maroon" title="c7">D1</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</font>) implies <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</font>,<font color="Maroon" title="c7">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>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T6" title="ENUMSET1:th.6">ENUMSET1:6</a></span>;<br/><a NAME="E13:63_1_1"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</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#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">C1</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>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:63_1_1"><span class="lab"><font color="Green" title="E59">A11</font></span></a>, <a class="ref" href="enumset1.html#T3" title="ENUMSET1:th.3">ENUMSET1:3</a></span>;<br/><a NAME="E14:63_1_1"/><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="c4">A1</font>,<font color="Maroon" title="c5">B1</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>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th21</a></span>;<br/><a NAME="E15:63_1_1"/><span class="kw">then </span>
 <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>) <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E2:63_1_1"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">Def14</a></span>;<br/><a NAME="E16:63_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E56">A8</font></span></a>, <a class="ref" href="incsp_1.html#D17" target="_self" title="INCSP_1:def.17">Def17</a></span>;<br/><a NAME="E17:63_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A13</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c6">C1</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>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:63_1_1"><span class="lab"><font color="Green" title="E58">A10</font></span></a>, <a class="ref" href="incsp_1.html#T4" target="_self" title="INCSP_1:th.4">Th14</a></span>;<br/><a NAME="E18:63_1_1"/>
<font color="Maroon" title="c4">A1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a>  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_1"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="incsp_1.html#T1" target="_self" title="INCSP_1:th.1">Th11</a></span>;<br/><a NAME="E19:63_1_1"/><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">A1</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>  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E56">A8</font></span></a>, <a class="ref" href="incsp_1.html#T1" target="_self" title="INCSP_1:th.1">Th11</a></span>;<br/><a NAME="E20:63_1_1"/><span class="kw">then </span>
 not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c6">C1</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E1:63_1"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="txt" href="#E7:63_1_1"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="txt" href="#E5:63_1_1"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="ref" href="incsp_1.html#D6" target="_self" title="INCSP_1:def.6">Def6</a>, <a class="ref" href="incsp_1.html#T18" target="_self" title="INCSP_1:th.18">Th39</a></span>;<br/><a NAME="E21:63_1_1"/><span class="kw">then </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">A1</font>,<font color="Maroon" title="c6">C1</font>,<font color="Maroon" title="c7">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E12:63_1_1"><span class="lab"><font color="Green" title="E60">A12</font></span></a>, <a class="txt" href="#E17:63_1_1"><span class="lab"><font color="Green" title="E61">A13</font></span></a>, <a class="ref" href="incsp_1.html#D7" target="_self" title="INCSP_1:def.7">Def7</a>, <a class="ref" href="incsp_1.html#T19" target="_self" title="INCSP_1:th.19">Th40</a></span>;<br/><span class="kw">hence </span><a NAME="E22:63_1_1"/>
 ex <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> st <br/>( <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp;  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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E49">A1</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><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c8">Q</font> =  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c3">A</font>);<br/><span class="kw">assume </span><a NAME="E1:63_1_2"/><span class="lab"><font color="Green" title="E53">A14</font></span>: 
 not <font color="Maroon" title="c3">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a>  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</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">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> st <br/>( <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp;  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/><a NAME="E2:63_1_2"/><span class="lab"><font color="Green" title="E54">A15</font></span>: 
<font color="Maroon" title="c4">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="incsp_1.html#T16" target="_self" title="INCSP_1:th.16">Th37</a></span>;<br/><a NAME="E3:63_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A16</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</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>  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</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="E4:63_1_2"/><span class="kw">then </span>
 not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c3">A</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2"><span class="lab"><font color="Green" title="E53">A14</font></span></a>, <a class="txt" href="#E2:63_1_2"><span class="lab"><font color="Green" title="E54">A15</font></span></a>, <a class="ref" href="incsp_1.html#T18" target="_self" title="INCSP_1:th.18">Th39</a></span>;<br/><a NAME="E5:63_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A17</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c3">A</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>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c3">A</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#D20" target="_self" title="INCSP_1:def.20">Def20</a></span>;<br/><a NAME="E6:63_1_2"/><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="c4">A1</font>,<font color="Maroon" title="c5">B1</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#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">A</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>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c3">A</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T3" title="ENUMSET1:th.3">ENUMSET1:3</a></span>;<br/><a NAME="E7:63_1_2"/><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="c4">A1</font>,<font color="Maroon" title="c5">B1</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>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c3">A</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T9" target="_self" title="INCSP_1:th.9">Th19</a></span>;<br/><a NAME="E8:63_1_2"/><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="c6">C1</font>,<font color="Maroon" title="c7">D1</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>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c3">A</font>) implies <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</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="c6">C1</font>,<font color="Maroon" title="c7">D1</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>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c3">A</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th21</a></span>;<br/><a NAME="E9:63_1_2"/><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="c6">C1</font>,<font color="Maroon" title="c7">D1</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>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c3">A</font>) implies <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">C1</font>,<font color="Maroon" title="c7">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>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c3">A</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/><a NAME="E10:63_1_2"/><span class="kw">then </span>
(  not <font color="Maroon" title="c6">C1</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c3">A</font>) or  not <font color="Maroon" title="c7">D1</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c3">A</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="incsp_1.html#D7" target="_self" title="INCSP_1:def.7">Def7</a>, <a class="ref" href="incsp_1.html#T3" target="_self" title="INCSP_1:th.3">Th13</a></span>;<br/><a NAME="E11:63_1_2"/><span class="kw">then </span>
(  not <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c6">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="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c7">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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2"><span class="lab"><font color="Green" title="E53">A14</font></span></a>, <a class="txt" href="#E2:63_1_2"><span class="lab"><font color="Green" title="E54">A15</font></span></a>, <a class="txt" href="#E3:63_1_2"><span class="lab"><font color="Green" title="E55">A16</font></span></a>, <a class="txt" href="#E5:63_1_2"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="ref" href="incsp_1.html#T18" target="_self" title="INCSP_1:th.18">Th39</a>, <a class="ref" href="incsp_1.html#T19" target="_self" title="INCSP_1:th.19">Th40</a></span>;<br/><a NAME="E12:63_1_2"/><span class="kw">then </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">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c6">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="c3">A</font>,<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c7">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>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T67" title="ENUMSET1:th.67">ENUMSET1:67</a></span>;<br/><span class="kw">hence </span><a NAME="E13:63_1_2"/>
 ex <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> st <br/>( <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp;  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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E49">A1</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><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:63_1"/>
 ex <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> st <br/>( <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp;  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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1"><span class="lab"><font color="Green" title="E52">A4</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:63"/>
 ex <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> st <br/>( <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp;  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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63"><span class="lab"><font color="Green" title="E50">A2</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/>


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