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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">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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>



<b>consider </b><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><b> such that </b><br/><a NAME="E2:64"/><i><font color="Green" title="E50">A1</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="incsp_1.html#D11" target="_self" title="INCSP_1:def.11">Def11</a></i>;<br/>
<b>consider </b><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><b> such that </b><br/><a NAME="E4:64"/><i><font color="Green" title="E51">A2</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="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>
 <b>by </b><i><a class="ref" href="incsp_1.html#D16" target="_self" title="INCSP_1:def.16">Def16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_1"/><i><font color="Green" title="E52">A3</font></i>: 
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:()"><i><font color="Red">::  thesis: </font></i></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><i><font color="Green" title="E53">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_1_1"/><i><font color="Green" title="E54">A5</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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/><b>set </b><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/><a NAME="E2:64_1_1"/><i><font color="Green" title="E55">A6</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E4:64"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="incsp_1.html#T37" target="_self" title="INCSP_1:th.37">Th37</a></i>;<br/><a NAME="E3:64_1_1"/><b>then </b><i><font color="Green" title="E56">A7</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">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>
 <b>by </b><i><a class="ref" href="incsp_1.html#D19" target="_self" title="INCSP_1:def.19">Def19</a></i>;<br/><a NAME="E4:64_1_1"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a></i>;<br/><a NAME="E5:64_1_1"/><b>then </b><i><font color="Green" title="E57">A8</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E1:64_1_1"><i><font color="Green" title="E54">A5</font></i></a>, <a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a></i>;<br/><a NAME="E6:64_1_1"/><i><font color="Green" title="E58">A9</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="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>
 <b>by </b><i><a class="txt" href="#E4:64"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="incsp_1.html#T38" target="_self" title="INCSP_1:th.38">Th38</a></i>;<br/><a NAME="E7:64_1_1"/><b>then </b><i><font color="Green" title="E59">A10</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="incsp_1.html#D20" target="_self" title="INCSP_1:def.20">Def20</a></i>;<br/><a NAME="E8:64_1_1"/><b>then </b>
( <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> )
 <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="E9:64_1_1"/><b>then </b><i><font color="Green" title="E60">A11</font></i>: 
( <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> )
 <b>by </b><i><a class="ref" href="enumset1.html#T46" title="ENUMSET1:th.46">ENUMSET1:46</a></i>;<br/><a NAME="E10:64_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>
 <b>by </b><i><a class="txt" href="#E2:64_1_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="ref" href="incsp_1.html#D19" target="_self" title="INCSP_1:def.19">Def19</a></i>;<br/><a NAME="E11:64_1_1"/><b>then </b>
( <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> )
 <b>by </b><i><a class="ref" href="incsp_1.html#T18" target="_self" title="INCSP_1:th.18">Th18</a></i>;<br/><a NAME="E12:64_1_1"/><b>then </b>
( <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> )
 <b>by </b><i><a class="ref" href="enumset1.html#T43" title="ENUMSET1:th.43">ENUMSET1:43</a></i>;<br/><a NAME="E13:64_1_1"/><b>then </b><i><font color="Green" title="E61">A12</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="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>
 <b>by </b><i><a class="txt" href="#E2:64"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="#E1:64_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="#E5:64_1_1"><i><font color="Green" title="E57">A8</font></i></a>, <a class="txt" href="#E6:64_1_1"><i><font color="Green" title="E58">A9</font></i></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#T39" target="_self" title="INCSP_1:th.39">Th39</a></i>;<br/><a NAME="E14:64_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>
 <b>by </b><i><a class="txt" href="#E7:64_1_1"><i><font color="Green" title="E59">A10</font></i></a>, <a class="ref" href="enumset1.html#T43" title="ENUMSET1:th.43">ENUMSET1:43</a></i>;<br/><a NAME="E15:64_1_1"/><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">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>
 <b>by </b><i><a class="ref" href="incsp_1.html#T21" target="_self" title="INCSP_1:th.21">Th21</a></i>;<br/><a NAME="E16:64_1_1"/><b>then </b>
(  <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="#E2:64_1_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="txt" href="#E3:64_1_1"><i><font color="Green" title="E56">A7</font></i></a>, <a class="txt" href="#E6:64_1_1"><i><font color="Green" title="E58">A9</font></i></a>, <a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">Def14</a>, <a class="ref" href="incsp_1.html#D20" target="_self" title="INCSP_1:def.20">Def20</a></i>;<br/><a NAME="E17:64_1_1"/><b>then </b>
( <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="#E1:64_1_1"><i><font color="Green" title="E54">A5</font></i></a>, <a class="ref" href="incsp_1.html#D17" target="_self" title="INCSP_1:def.17">Def17</a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></i>;<br/><a NAME="E18:64_1_1"/><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="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>
 <b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></i>;<br/><a NAME="E19:64_1_1"/><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="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>
 <b>by </b><i><a class="txt" href="#E4:64"><i><font color="Green" title="E51">A2</font></i></a>, <a class="txt" href="#E9:64_1_1"><i><font color="Green" title="E60">A11</font></i></a>, <a class="txt" href="#E13:64_1_1"><i><font color="Green" title="E61">A12</font></i></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#T40" target="_self" title="INCSP_1:th.40">Th40</a></i>;<br/><b>hence </b><a NAME="E20:64_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> )
 <b>by </b><i><a class="txt" href="#E2:64"><i><font color="Green" title="E50">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/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_1_2"/><i><font color="Green" title="E54">A13</font></i>: 
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:()"><i><font color="Red">::  thesis: </font></i></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/><b>set </b><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/><a NAME="E2:64_1_2"/><i><font color="Green" title="E55">A14</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E4:64"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="incsp_1.html#T37" target="_self" title="INCSP_1:th.37">Th37</a></i>;<br/><a NAME="E3:64_1_2"/><b>then </b><i><font color="Green" title="E56">A15</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">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>
 <b>by </b><i><a class="ref" href="incsp_1.html#D19" target="_self" title="INCSP_1:def.19">Def19</a></i>;<br/><a NAME="E4:64_1_2"/><b>then </b>
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>
 <b>by </b><i><a class="txt" href="#E1:64_1_2"><i><font color="Green" title="E54">A13</font></i></a>, <a class="txt" href="#E2:64_1_2"><i><font color="Green" title="E55">A14</font></i></a>, <a class="ref" href="incsp_1.html#T39" target="_self" title="INCSP_1:th.39">Th39</a></i>;<br/><a NAME="E5:64_1_2"/><b>then </b><i><font color="Green" title="E57">A16</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="incsp_1.html#D20" target="_self" title="INCSP_1:def.20">Def20</a></i>;<br/><a NAME="E6:64_1_2"/><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">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>
 <b>by </b><i><a class="ref" href="enumset1.html#T43" title="ENUMSET1:th.43">ENUMSET1:43</a></i>;<br/><a NAME="E7:64_1_2"/><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">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>
 <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="E8:64_1_2"/><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="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> )
 <b>by </b><i><a class="ref" href="incsp_1.html#T21" target="_self" title="INCSP_1:th.21">Th21</a></i>;<br/><a NAME="E9:64_1_2"/><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="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> )
 <b>by </b><i><a class="ref" href="enumset1.html#T45" title="ENUMSET1:th.45">ENUMSET1:45</a></i>;<br/><a NAME="E10:64_1_2"/><b>then </b>
(  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> )
 <b>by </b><i><a class="txt" href="#E4:64"><i><font color="Green" title="E51">A2</font></i></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#T13" target="_self" title="INCSP_1:th.13">Th13</a></i>;<br/><a NAME="E11:64_1_2"/><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="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> )
 <b>by </b><i><a class="txt" href="#E1:64_1_2"><i><font color="Green" title="E54">A13</font></i></a>, <a class="txt" href="#E2:64_1_2"><i><font color="Green" title="E55">A14</font></i></a>, <a class="txt" href="#E3:64_1_2"><i><font color="Green" title="E56">A15</font></i></a>, <a class="txt" href="#E5:64_1_2"><i><font color="Green" title="E57">A16</font></i></a>, <a class="ref" href="incsp_1.html#T39" target="_self" title="INCSP_1:th.39">Th39</a>, <a class="ref" href="incsp_1.html#T40" target="_self" title="INCSP_1:th.40">Th40</a></i>;<br/><a NAME="E12:64_1_2"/><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="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> )
 <b>by </b><i><a class="ref" href="enumset1.html#T110" title="ENUMSET1:th.110">ENUMSET1:110</a></i>;<br/><b>hence </b><a NAME="E13:64_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> )
 <b>by </b><i><a class="txt" href="#E2:64"><i><font color="Green" title="E50">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/></div><b>end;</b></div><b>hence </b><a NAME="E4:64_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> )
 <b>by </b><i><a class="txt" href="#E2:64_1"><i><font color="Green" title="E53">A4</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/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:64"/>
 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> )
 <b>by </b><i><a class="txt" href="#E4:64"><i><font color="Green" title="E51">A2</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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