<?xml version="1.0"?>
<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="Red">::  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> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st <br/>( <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</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></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 <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</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></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><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="Red">::  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> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st <br/>( <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="Maroon" title="c2">P</font> &amp; 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>  )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c3">A1</font>, <font color="Maroon" title="c4">B1</font>, <font color="Maroon" title="c5">C1</font>, <font color="Maroon" title="c6">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="E2:94"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
<font color="Maroon" title="c3">A1</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">and </span><br/><a NAME="E3:94"/><span class="lab"><font color="Green" title="E52">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="c3">A1</font>,<font color="Maroon" title="c4">B1</font>,<font color="Maroon" title="c5">C1</font>,<font color="Maroon" title="c6">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="incsp_1.html#E94"><span class="lab"><font color="Green" title="E49">Lm2</font></span></a></span>;<br/>
<a NAME="E4:94"/>
not <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c4">B1</font>,<font color="Maroon" title="c6">D1</font>,<font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c5">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:94"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="enumset1.html#T112" title="ENUMSET1:th.112">ENUMSET1:112</a></span>;<br/>
<a NAME="E5:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A3</font></span>: 
<font color="Maroon" title="c4">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T37" target="_self" title="INCSP_1:th.37">Th37</a></span>;<br/>
<a NAME="E6:94"/>
not <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c5">C1</font>,<font color="Maroon" title="c6">D1</font>,<font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c4">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:94"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="enumset1.html#T118" title="ENUMSET1:th.118">ENUMSET1:118</a></span>;<br/>
<a NAME="E7:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A4</font></span>: 
<font color="Maroon" title="c5">C1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T37" target="_self" title="INCSP_1:th.37">Th37</a></span>;<br/>
<a NAME="E8:94"/>
not <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c4">B1</font>,<font color="Maroon" title="c5">C1</font>,<font color="Maroon" title="c6">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="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:94"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="incsp_1.html#D7" target="_self" title="INCSP_1:def.7">Def7</a></span>;<br/>
<a NAME="E9:94"/><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">B1</font>,<font color="Maroon" title="c5">C1</font>,<font color="Maroon" title="c6">D1</font>,<font color="Maroon" title="c3">A1</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="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T111" title="ENUMSET1:th.111">ENUMSET1:111</a></span>;<br/>
<a NAME="E10:94"/><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">B1</font>,<font color="Maroon" title="c5">C1</font>,<font color="Maroon" title="c6">D1</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="c3">A1</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="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T46" title="ENUMSET1:th.46">ENUMSET1:46</a></span>;<br/>
<a NAME="E11:94"/><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">B1</font>,<font color="Maroon" title="c5">C1</font>,<font color="Maroon" title="c6">D1</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="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:94"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="incsp_1.html#T19" target="_self" title="INCSP_1:th.19">Th19</a></span>;<br/>
<a NAME="E12:94"/><span class="kw">then </span>
(  not <font color="Maroon" title="c4">B1</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> or  not <font color="Maroon" title="c5">C1</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> or  not <font color="Maroon" title="c6">D1</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#T14" target="_self" title="INCSP_1:th.14">Th14</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">X</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="E14:94"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
( <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B1</font> or <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C1</font> or <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D1</font> )
 <span class="kw">and </span><br/><a NAME="E15:94"/><span class="lab"><font color="Green" title="E56">A6</font></span>: 
not <font color="Maroon" title="c7">X</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font>
 ;<br/>
<a NAME="E16:94"/>
not <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c4">B1</font>,<font color="Maroon" title="c5">C1</font>,<font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c6">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="#E3:94"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="enumset1.html#T110" title="ENUMSET1:th.110">ENUMSET1:110</a></span>;<br/>
<a NAME="E17:94"/><span class="kw">then </span>
<font color="Maroon" title="c4">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T37" target="_self" title="INCSP_1:th.37">Th37</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">Y</font>, <font color="Maroon" title="c9">Z</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="E19:94"/><span class="lab"><font color="Green" title="E57">A7</font></span>: 
( ( <font color="Maroon" title="c8">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B1</font> or <font color="Maroon" title="c8">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C1</font> or <font color="Maroon" title="c8">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D1</font> ) &amp; ( <font color="Maroon" title="c9">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B1</font> or <font color="Maroon" title="c9">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C1</font> or <font color="Maroon" title="c9">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D1</font> ) &amp; <font color="Maroon" title="c8">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Maroon" title="c9">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Maroon" title="c8">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">Z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:94"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="#E5:94"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E7:94"><span class="lab"><font color="Green" title="E54">A4</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">P1</font> =  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c3">A1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">P2</font> =  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c9">Z</font>,<font color="Maroon" title="c3">A1</font>;<br/>
<div><span class="lab"><font color="Green" title="E58">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:94_1"/>
<span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c9">Z</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:94_1"/><span class="kw">then </span>
( <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c6">D1</font>,<font color="Maroon" title="c4">B1</font>,<font color="Maroon" title="c5">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 <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c6">D1</font>,<font color="Maroon" title="c5">C1</font>,<font color="Maroon" title="c4">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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:94"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E14:94"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="#E19:94"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="enumset1.html#T104" title="ENUMSET1:th.104">ENUMSET1:104</a></span>;<br/><span class="kw">hence </span><a NAME="E3:94_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:94"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="enumset1.html#T105" title="ENUMSET1:th.105">ENUMSET1:105</a>, <a class="ref" href="enumset1.html#T107" title="ENUMSET1:th.107">ENUMSET1:107</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E21:94"/><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">A1</font>,<font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">Y</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="ref" href="incsp_1.html#T38" target="_self" title="INCSP_1:th.38">Th38</a></span>;<br/>
<a NAME="E22:94"/><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="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c3">A1</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="ref" href="enumset1.html#T100" title="ENUMSET1:th.100">ENUMSET1:100</a></span>;<br/>
<a NAME="E23:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A9</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c3">A1</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="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c3">A1</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="E24:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A10</font></span>: 
<font color="Maroon" title="c3">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="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c3">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">B</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="E26:94"/><span class="lab"><font color="Green" title="E61">A11</font></span>: 
<font color="Maroon" title="c3">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">B</font>
 <span class="kw">and </span><br/><a NAME="E27:94"/><span class="lab"><font color="Green" title="E62">A12</font></span>: 
<font color="Maroon" title="c12">B</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="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c3">A1</font>
 <span class="kw">and </span><br/><a NAME="E28:94"/><span class="lab"><font color="Green" title="E63">A13</font></span>: 
<font color="Maroon" title="c12">B</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="txt" href="#E2:94"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="incsp_1.html#D15" target="_self" title="INCSP_1:def.15">Def15</a></span>;<br/>
<a NAME="E29:94"/>
not <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c9">Z</font>,<font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c8">Y</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="#E20:94"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="ref" href="enumset1.html#T112" title="ENUMSET1:th.112">ENUMSET1:112</a></span>;<br/>
<a NAME="E30:94"/><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="c7">X</font>,<font color="Maroon" title="c9">Z</font>,<font color="Maroon" title="c3">A1</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="ref" href="incsp_1.html#T38" target="_self" title="INCSP_1:th.38">Th38</a></span>;<br/>
<a NAME="E31:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A14</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c9">Z</font>,<font color="Maroon" title="c3">A1</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="c7">X</font>,<font color="Maroon" title="c9">Z</font>,<font color="Maroon" title="c3">A1</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="E32:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A15</font></span>: 
<font color="Maroon" title="c3">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="c7">X</font>,<font color="Maroon" title="c9">Z</font>,<font color="Maroon" title="c3">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">C</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="E34:94"/><span class="lab"><font color="Green" title="E66">A16</font></span>: 
<font color="Maroon" title="c3">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">C</font>
 <span class="kw">and </span><br/><a NAME="E35:94"/><span class="lab"><font color="Green" title="E67">A17</font></span>: 
<font color="Maroon" title="c13">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">and </span><br/><a NAME="E36:94"/><span class="lab"><font color="Green" title="E68">A18</font></span>: 
<font color="Maroon" title="c13">C</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="c7">X</font>,<font color="Maroon" title="c9">Z</font>,<font color="Maroon" title="c3">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:94"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="incsp_1.html#D15" target="_self" title="INCSP_1:def.15">Def15</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">A1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">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 <br/>( <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">A1</font>,<font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">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="c2">P</font> &amp; not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">A1</font>,<font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">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>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c12">B</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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 <br/>( <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c12">B</font>,<font color="Olive" title="b1">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="c2">P</font> &amp; not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c12">B</font>,<font color="Olive" title="b1">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>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c13">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">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="c2">P</font> &amp; not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">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>  )</span><br/>

<span class="kw">thus </span><a NAME="E37:94"/>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:94"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E28:94"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="txt" href="#E35:94"><span class="lab"><font color="Green" title="E67">A17</font></span></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c3">A1</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">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> </span><br/>

<span class="kw">given </span><font color="Maroon" title="c14">K</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><a NAME="E39:94"/><span class="lab"><font color="Green" title="E69">A19</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">A1</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c14">K</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="incsp_1.html#D6" target="_self" title="INCSP_1:def.6">INCSP_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E40:94"/><span class="lab"><font color="Green" title="E70">A20</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c13">C</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="c7">X</font>,<font color="Maroon" title="c9">Z</font>,<font color="Maroon" title="c3">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:94"><span class="lab"><font color="Green" title="E65">A15</font></span></a>, <a class="txt" href="#E36:94"><span class="lab"><font color="Green" title="E68">A18</font></span></a>, <a class="ref" href="incsp_1.html#T13" target="_self" title="INCSP_1:th.13">Th13</a></span>;<br/>
<a NAME="E41:94"/>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c13">C</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c14">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:94"><span class="lab"><font color="Green" title="E69">A19</font></span></a>, <a class="ref" href="enumset1.html#T98" title="ENUMSET1:th.98">ENUMSET1:98</a></span>;<br/>
<a NAME="E42:94"/><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">A1</font>,<font color="Maroon" title="c13">C</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="c12">B</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> <font color="Maroon" title="c14">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T43" title="ENUMSET1:th.43">ENUMSET1:43</a></span>;<br/>
<a NAME="E43:94"/><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">A1</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c14">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T20" target="_self" title="INCSP_1:th.20">Th20</a></span>;<br/>
<a NAME="E44:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A21</font></span>: 
<font color="Maroon" title="c14">K</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="c7">X</font>,<font color="Maroon" title="c9">Z</font>,<font color="Maroon" title="c3">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:94"><span class="lab"><font color="Green" title="E66">A16</font></span></a>, <a class="txt" href="#E40:94"><span class="lab"><font color="Green" title="E70">A20</font></span></a>, <a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">Def14</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">E</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="E46:94"/><span class="lab"><font color="Green" title="E72">A22</font></span>: 
<font color="Maroon" title="c12">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">E</font>
 <span class="kw">and </span><br/><a NAME="E47:94"/><span class="lab"><font color="Green" title="E73">A23</font></span>: 
<font color="Maroon" title="c15">E</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="incsp_1.html#E60"><span class="lab"><font color="Green" title="E36">Lm1</font></span></a></span>;<br/>
<a NAME="E48:94"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c12">B</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="c13">C</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> <font color="Maroon" title="c14">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:94"><span class="lab"><font color="Green" title="E69">A19</font></span></a>, <a class="ref" href="enumset1.html#T43" title="ENUMSET1:th.43">ENUMSET1:43</a></span>;<br/>
<a NAME="E49:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A24</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c14">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T20" target="_self" title="INCSP_1:th.20">Th20</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E75">A25</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="94_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:94_2"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:94"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E28:94"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="incsp_1.html#T13" target="_self" title="INCSP_1:th.13">Th13</a></span>;<br/><a NAME="E2:94_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:94"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="txt" href="#E49:94"><span class="lab"><font color="Green" title="E74">A24</font></span></a>, <a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">Def14</a></span>;<br/><a NAME="E3:94_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">E</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="txt" href="#E47:94"><span class="lab"><font color="Green" title="E73">A23</font></span></a>, <a class="ref" href="incsp_1.html#D17" target="_self" title="INCSP_1:def.17">Def17</a></span>;<br/><a NAME="E4:94_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A26</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c15">E</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:94"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="incsp_1.html#T13" target="_self" title="INCSP_1:th.13">Th13</a></span>;<br/><span class="kw">assume </span><a NAME="E5:94_2"/>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c15">E</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">L</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E7:94_2"/><span class="lab"><font color="Green" title="E77">A27</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c15">E</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c16">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#D6" target="_self" title="INCSP_1:def.6">Def6</a></span>;<br/><a NAME="E8:94_2"/><span class="lab"><font color="Green" title="E78">A28</font></span>: 
<font color="Maroon" title="c7">X</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:94_2"><span class="lab"><font color="Green" title="E77">A27</font></span></a>, <a class="ref" href="incsp_1.html#T12" target="_self" title="INCSP_1:th.12">Th12</a></span>;<br/><a NAME="E9:94_2"/>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c15">E</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c7">X</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c16">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:94_2"><span class="lab"><font color="Green" title="E77">A27</font></span></a>, <a class="ref" href="enumset1.html#T102" title="ENUMSET1:th.102">ENUMSET1:102</a></span>;<br/><a NAME="E10:94_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="c15">E</font>,<font color="Maroon" title="c12">B</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="c7">X</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> <font color="Maroon" title="c16">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T43" title="ENUMSET1:th.43">ENUMSET1:43</a></span>;<br/><a NAME="E11:94_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="c15">E</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c16">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T18" target="_self" title="INCSP_1:th.18">Th18</a></span>;<br/><a NAME="E12:94_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:94"><span class="lab"><font color="Green" title="E72">A22</font></span></a>, <a class="txt" href="#E4:94_2"><span class="lab"><font color="Green" title="E76">A26</font></span></a>, <a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">Def14</a></span>;<br/><span class="kw">hence </span><a NAME="E13:94_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:94"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="txt" href="#E8:94_2"><span class="lab"><font color="Green" title="E78">A28</font></span></a>, <a class="ref" href="incsp_1.html#D17" target="_self" title="INCSP_1:def.17">Def17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E51:94"/>
<font color="Maroon" title="c12">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:94"><span class="lab"><font color="Green" title="E69">A19</font></span></a>, <a class="ref" href="incsp_1.html#T12" target="_self" title="INCSP_1:th.12">Th12</a></span>;<br/>
<a NAME="E52:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A29</font></span>: 
<font color="Maroon" title="c12">B</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="c7">X</font>,<font color="Maroon" title="c9">Z</font>,<font color="Maroon" title="c3">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:94"><span class="lab"><font color="Green" title="E71">A21</font></span></a>, <a class="ref" href="incsp_1.html#D17" target="_self" title="INCSP_1:def.17">Def17</a></span>;<br/>
<a NAME="E53:94"/><span class="lab"><font color="Green" title="E77">A30</font></span>: 
<font color="Maroon" title="c7">X</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="c7">X</font>,<font color="Maroon" title="c9">Z</font>,<font color="Maroon" title="c3">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:94"><span class="lab"><font color="Green" title="E64">A14</font></span></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></span>;<br/>
<a NAME="E54:94"/><span class="lab"><font color="Green" title="E78">A31</font></span>: 
<font color="Maroon" title="c7">X</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="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c3">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:94"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></span>;<br/>
<a NAME="E55:94"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c3">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:94"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="txt" href="#E27:94"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="ref" href="incsp_1.html#T13" target="_self" title="INCSP_1:th.13">Th13</a></span>;<br/>
<a NAME="E56:94"/><span class="kw">then </span>
<font color="Maroon" title="c14">K</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="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c3">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:94"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="txt" href="#E49:94"><span class="lab"><font color="Green" title="E74">A24</font></span></a>, <a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">Def14</a></span>;<br/>
<a NAME="E57:94"/><span class="kw">then </span>
<font color="Maroon" title="c15">E</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="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c3">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:94"><span class="lab"><font color="Green" title="E73">A23</font></span></a>, <a class="ref" href="incsp_1.html#D17" target="_self" title="INCSP_1:def.17">Def17</a></span>;<br/>
<a NAME="E58:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A32</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c15">E</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="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c3">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:94"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="txt" href="#E54:94"><span class="lab"><font color="Green" title="E78">A31</font></span></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></span>;<br/>
<a NAME="E59:94"/>
<font color="Maroon" title="c15">E</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="c7">X</font>,<font color="Maroon" title="c9">Z</font>,<font color="Maroon" title="c3">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:94"><span class="lab"><font color="Green" title="E73">A23</font></span></a>, <a class="txt" href="#E44:94"><span class="lab"><font color="Green" title="E71">A21</font></span></a>, <a class="ref" href="incsp_1.html#D17" target="_self" title="INCSP_1:def.17">Def17</a></span>;<br/>
<a NAME="E60:94"/><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="c7">X</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c15">E</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="c7">X</font>,<font color="Maroon" title="c9">Z</font>,<font color="Maroon" title="c3">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:94"><span class="lab"><font color="Green" title="E76">A29</font></span></a>, <a class="txt" href="#E53:94"><span class="lab"><font color="Green" title="E77">A30</font></span></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></span>;<br/>
<a NAME="E61:94"/><span class="kw">then </span>
 <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c9">Z</font>,<font color="Maroon" title="c3">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:94"><span class="lab"><font color="Green" title="E75">A25</font></span></a>, <a class="txt" href="#E58:94"><span class="lab"><font color="Green" title="E79">A32</font></span></a>, <a class="ref" href="incsp_1.html#D13" target="_self" title="INCSP_1:def.13">Def13</a></span>;<br/>
<a NAME="E62:94"/><span class="kw">then </span>
<font color="Maroon" title="c9">Z</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="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c3">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:94"><span class="lab"><font color="Green" title="E64">A14</font></span></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></span>;<br/>
<a NAME="E63:94"/><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="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c3">A1</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="c9">Z</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="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c3">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:94"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="incsp_1.html#T19" target="_self" title="INCSP_1:th.19">Th19</a></span>;<br/>
<a NAME="E64:94"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c9">Z</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="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c3">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T46" title="ENUMSET1:th.46">ENUMSET1:46</a></span>;<br/>
<a NAME="E65:94"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c9">Z</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#D7" target="_self" title="INCSP_1:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E66:94"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:94"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="ref" href="enumset1.html#T110" title="ENUMSET1:th.110">ENUMSET1:110</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
