<?xml version="1.0"?>
<div class="add">

<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> 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/><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> 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/>



<b>consider </b><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><b> such that </b><br/><a NAME="E2:65"/><i><font color="Green" title="E51">A1</font></i>: 
<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>
 <b>and </b><br/><a NAME="E3:65"/><i><font color="Green" title="E52">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="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>
 <b>by </b><i><a class="txt" href="incsp_1.html#E94"><i><font color="Green" title="E49">Lm2</font></i></a></i>;<br/>
<a NAME="E4:65"/>
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>
 
<b>by </b><i><a class="txt" href="#E3:65"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="incsp_1.html#D7" target="_self" title="INCSP_1:def.7">Def7</a></i>;<br/>
<a NAME="E5:65"/><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">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>
 
<b>by </b><i><a class="ref" href="enumset1.html#T111" title="ENUMSET1:th.111">ENUMSET1:111</a></i>;<br/>
<a NAME="E6:65"/><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">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>
 
<b>by </b><i><a class="ref" href="enumset1.html#T46" title="ENUMSET1:th.46">ENUMSET1:46</a></i>;<br/>
<a NAME="E7:65"/><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">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>
 
<b>by </b><i><a class="txt" href="#E2:65"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="incsp_1.html#T19" target="_self" title="INCSP_1:th.19">Th19</a></i>;<br/>
<a NAME="E8:65"/><b>then </b>
(  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> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></i>;<br/>
<b>then consider </b><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><b> such that </b><br/><a NAME="E10:65"/><i><font color="Green" title="E53">A3</font></i>: 
( <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> )
 <b>and </b><br/><a NAME="E11:65"/><i><font color="Green" title="E54">A4</font></i>: 
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="E12:65"/>
( 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> &amp; 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> &amp; 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> )
 
<b>by </b><i><a class="txt" href="#E3:65"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="enumset1.html#T110" title="ENUMSET1:th.110">ENUMSET1:110</a>, <a class="ref" href="enumset1.html#T112" title="ENUMSET1:th.112">ENUMSET1:112</a>, <a class="ref" href="enumset1.html#T118" title="ENUMSET1:th.118">ENUMSET1:118</a></i>;<br/>
<a NAME="E13:65"/><b>then </b>
( <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> &amp; <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> &amp; <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> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T37" target="_self" title="INCSP_1:th.37">Th37</a></i>;<br/>
<b>then consider </b><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><b> such that </b><br/><a NAME="E15:65"/><i><font color="Green" title="E55">A5</font></i>: 
( ( <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> )
 <b>by </b><i><a class="txt" href="#E10:65"><i><font color="Green" title="E53">A3</font></i></a></i>;<br/>
<div><i><font color="Green" title="E56">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:65_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:65_1"/><b>then </b>
( <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> )
 <b>by </b><i><a class="txt" href="#E3:65"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="#E10:65"><i><font color="Green" title="E53">A3</font></i></a>, <a class="txt" href="#E15:65"><i><font color="Green" title="E55">A5</font></i></a>, <a class="ref" href="enumset1.html#T104" title="ENUMSET1:th.104">ENUMSET1:104</a></i>;<br/><b>hence </b><a NAME="E3:65_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:65"><i><font color="Green" title="E52">A2</font></i></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></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>set </b><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/>
<b>set </b><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/>
<a NAME="E17:65"/>
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>
 
<b>by </b><i><a class="txt" href="#E16:65"><i><font color="Green" title="E56">A6</font></i></a>, <a class="ref" href="incsp_1.html#T38" target="_self" title="INCSP_1:th.38">Th38</a></i>;<br/>
<a NAME="E18:65"/><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="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>
 
<b>by </b><i><a class="ref" href="enumset1.html#T100" title="ENUMSET1:th.100">ENUMSET1:100</a></i>;<br/>
<a NAME="E19:65"/><b>then </b><i><font color="Green" title="E57">A7</font></i>: 
<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>
 
<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="E20:65"/><b>then </b><i><font color="Green" title="E58">A8</font></i>: 
<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>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></i>;<br/>
<b>then consider </b><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><b> such that </b><br/><a NAME="E22:65"/><i><font color="Green" title="E59">A9</font></i>: 
<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>
 <b>and </b><br/><a NAME="E23:65"/><i><font color="Green" title="E60">A10</font></i>: 
<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>
 <b>and </b><br/><a NAME="E24:65"/><i><font color="Green" title="E61">A11</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E2:65"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="incsp_1.html#D15" target="_self" title="INCSP_1:def.15">Def15</a></i>;<br/>
<a NAME="E25:65"/>
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>
 
<b>by </b><i><a class="txt" href="#E16:65"><i><font color="Green" title="E56">A6</font></i></a>, <a class="ref" href="enumset1.html#T112" title="ENUMSET1:th.112">ENUMSET1:112</a></i>;<br/>
<a NAME="E26:65"/><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="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>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T38" target="_self" title="INCSP_1:th.38">Th38</a></i>;<br/>
<a NAME="E27:65"/><b>then </b><i><font color="Green" title="E62">A12</font></i>: 
<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>
 
<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="E28:65"/><b>then </b><i><font color="Green" title="E63">A13</font></i>: 
<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>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></i>;<br/>
<b>then consider </b><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><b> such that </b><br/><a NAME="E30:65"/><i><font color="Green" title="E64">A14</font></i>: 
<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>
 <b>and </b><br/><a NAME="E31:65"/><i><font color="Green" title="E65">A15</font></i>: 
<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>
 <b>and </b><br/><a NAME="E32:65"/><i><font color="Green" title="E66">A16</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E2:65"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="incsp_1.html#D15" target="_self" title="INCSP_1:def.15">Def15</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">A1</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">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/>

<b>take </b>
<font color="Maroon" title="c12">B</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">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/>

<b>take </b>
<font color="Maroon" title="c13">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><a NAME="E33:65"/>
<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>
 <b>by </b><i><a class="txt" href="#E2:65"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E24:65"><i><font color="Green" title="E61">A11</font></i></a>, <a class="txt" href="#E31:65"><i><font color="Green" title="E65">A15</font></i></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="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/>

<b>given </b><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><b> such that </b><a NAME="E35:65"/><i><font color="Green" title="E67">A17</font></i>: 
<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>
 ; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E36:65"/><i><font color="Green" title="E68">A18</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="#E35:65"><i><font color="Green" title="E67">A17</font></i></a>, <a class="ref" href="incsp_1.html#T12" target="_self" title="INCSP_1:th.12">Th12</a></i>;<br/>
<b>consider </b><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><b> such that </b><br/><a NAME="E38:65"/><i><font color="Green" title="E69">A19</font></i>: 
<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>
 <b>and </b><br/><a NAME="E39:65"/><i><font color="Green" title="E70">A20</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="incsp_1.html#E60"><i><font color="Green" title="E36">Lm1</font></i></a></i>;<br/>
<a NAME="E40:65"/>
<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>
 
<b>by </b><i><a class="txt" href="#E35:65"><i><font color="Green" title="E67">A17</font></i></a>, <a class="ref" href="enumset1.html#T98" title="ENUMSET1:th.98">ENUMSET1:98</a></i>;<br/>
<a NAME="E41:65"/><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="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> &amp; <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> )
 
<b>by </b><i><a class="txt" href="#E35:65"><i><font color="Green" title="E67">A17</font></i></a>, <a class="ref" href="enumset1.html#T43" title="ENUMSET1:th.43">ENUMSET1:43</a></i>;<br/>
<a NAME="E42:65"/><b>then </b><i><font color="Green" title="E71">A21</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">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> &amp; <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> &amp; <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> &amp; <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> )
 
<b>by </b><i><a class="txt" href="#E20:65"><i><font color="Green" title="E58">A8</font></i></a>, <a class="txt" href="#E23:65"><i><font color="Green" title="E60">A10</font></i></a>, <a class="txt" href="#E28:65"><i><font color="Green" title="E63">A13</font></i></a>, <a class="txt" href="#E32:65"><i><font color="Green" title="E66">A16</font></i></a>, <a class="ref" href="incsp_1.html#T13" target="_self" title="INCSP_1:th.13">Th13</a>, <a class="ref" href="incsp_1.html#T20" target="_self" title="INCSP_1:th.20">Th20</a></i>;<br/>
<a NAME="E43:65"/><b>then </b>
( <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> &amp; <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> )
 
<b>by </b><i><a class="txt" href="#E22:65"><i><font color="Green" title="E59">A9</font></i></a>, <a class="txt" href="#E30:65"><i><font color="Green" title="E64">A14</font></i></a>, <a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">Def14</a></i>;<br/>
<a NAME="E44:65"/><b>then </b><i><font color="Green" title="E72">A22</font></i>: 
( <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> &amp; <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> &amp; <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> )
 
<b>by </b><i><a class="txt" href="#E36:65"><i><font color="Green" title="E68">A18</font></i></a>, <a class="txt" href="#E39:65"><i><font color="Green" title="E70">A20</font></i></a>, <a class="ref" href="incsp_1.html#D17" target="_self" title="INCSP_1:def.17">Def17</a></i>;<br/>
<div><i><font color="Green" title="E73">A23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:65_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><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><b> such that </b><br/><a NAME="E3:65_2"/><i><font color="Green" title="E74">A24</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="incsp_1.html#D6" target="_self" title="INCSP_1:def.6">Def6</a></i>;<br/><a NAME="E4:65_2"/><i><font color="Green" title="E75">A25</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E3:65_2"><i><font color="Green" title="E74">A24</font></i></a>, <a class="ref" href="enumset1.html#T102" title="ENUMSET1:th.102">ENUMSET1:102</a></i>;<br/><a NAME="E5:65_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>
 <b>by </b><i><a class="txt" href="#E2:65"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E24:65"><i><font color="Green" title="E61">A11</font></i></a>, <a class="ref" href="incsp_1.html#T13" target="_self" title="INCSP_1:th.13">Th13</a></i>;<br/><a NAME="E6:65_2"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E22:65"><i><font color="Green" title="E59">A9</font></i></a>, <a class="txt" href="#E42:65"><i><font color="Green" title="E71">A21</font></i></a>, <a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">Def14</a></i>;<br/><a NAME="E7:65_2"/><b>then </b>
( <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="#E39:65"><i><font color="Green" title="E70">A20</font></i></a>, <a class="txt" href="#E4:65_2"><i><font color="Green" title="E75">A25</font></i></a>, <a class="ref" href="incsp_1.html#D17" target="_self" title="INCSP_1:def.17">Def17</a>, <a class="ref" href="enumset1.html#T43" title="ENUMSET1:th.43">ENUMSET1:43</a></i>;<br/><a NAME="E8:65_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="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> &amp; <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> )
 <b>by </b><i><a class="txt" href="#E24:65"><i><font color="Green" title="E61">A11</font></i></a>, <a class="ref" href="incsp_1.html#T13" target="_self" title="INCSP_1:th.13">Th13</a>, <a class="ref" href="incsp_1.html#T18" target="_self" title="INCSP_1:th.18">Th18</a></i>;<br/><a NAME="E9:65_2"/><b>then </b>
( <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="#E38:65"><i><font color="Green" title="E69">A19</font></i></a>, <a class="txt" href="#E3:65_2"><i><font color="Green" title="E74">A24</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#T12" target="_self" title="INCSP_1:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E10:65_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E11:65"><i><font color="Green" title="E54">A4</font></i></a>, <a class="ref" href="incsp_1.html#D17" target="_self" title="INCSP_1:def.17">Def17</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>
<a NAME="E46:65"/>
( <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> &amp; <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> )
 
<b>by </b><i><a class="txt" href="#E19:65"><i><font color="Green" title="E57">A7</font></i></a>, <a class="txt" href="#E27:65"><i><font color="Green" title="E62">A12</font></i></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></i>;<br/>
<a NAME="E47:65"/><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="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> &amp; <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> )
 
<b>by </b><i><a class="txt" href="#E23:65"><i><font color="Green" title="E60">A10</font></i></a>, <a class="txt" href="#E44:65"><i><font color="Green" title="E72">A22</font></i></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></i>;<br/>
<a NAME="E48:65"/><b>then </b>
 <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>
 
<b>by </b><i><a class="txt" href="#E45:65"><i><font color="Green" title="E73">A23</font></i></a>, <a class="ref" href="incsp_1.html#D13" target="_self" title="INCSP_1:def.13">Def13</a></i>;<br/>
<a NAME="E49:65"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E27:65"><i><font color="Green" title="E62">A12</font></i></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></i>;<br/>
<a NAME="E50:65"/><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="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>
 
<b>by </b><i><a class="txt" href="#E19:65"><i><font color="Green" title="E57">A7</font></i></a>, <a class="ref" href="incsp_1.html#T19" target="_self" title="INCSP_1:th.19">Th19</a></i>;<br/>
<a NAME="E51:65"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="enumset1.html#T46" title="ENUMSET1:th.46">ENUMSET1:46</a></i>;<br/>
<a NAME="E52:65"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="incsp_1.html#D7" target="_self" title="INCSP_1:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E53:65"/>
contradiction
 <b>by </b><i><a class="txt" href="#E16:65"><i><font color="Green" title="E56">A6</font></i></a>, <a class="ref" href="enumset1.html#T110" title="ENUMSET1:th.110">ENUMSET1:110</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>
