<?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>;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font>;<br/>



<b>consider </b><font color="Maroon" title="c3">P</font> being   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E2:72"/><i><font color="Green" title="E46">A1</font></i>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D8" target="_self" title="INCSP_1:def.8">Def8</a></i>;<br/>
<a NAME="E3:72"/>
<font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="incsp_1.html#E2:72"><i><font color="Green" title="E46">A1</font></i></a>, <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="c4">L</font>, <font color="Maroon" title="c5">L1</font>, <font color="Maroon" title="c6">L2</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="E5:72"/><i><font color="Green" title="E47">A2</font></i>: 
<font color="Maroon" title="c5">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">L2</font>
 <b>and </b><br/><a NAME="E6:72"/><i><font color="Green" title="E48">A3</font></i>: 
( <font color="Maroon" title="c5">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c6">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp; not <font color="Maroon" title="c4">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> )
 <b>and </b><br/><a NAME="E7:72"/><i><font color="Green" title="E49">A4</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">L1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L2</font> )
 <b>by </b><i><a class="ref" href="incsp_1.html#T80" target="_self" title="INCSP_1:th.80">Th80</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">L</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">L1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c6">L2</font>
;<br/>

<b>thus </b><a NAME="E8:72"/>
( <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">L1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L2</font> )
 <b>by </b><i><a class="txt" href="incsp_1.html#E7:72"><i><font color="Green" title="E49">A4</font></i></a></i>;<br/>

<b>given </b><font color="Maroon" title="c7">Q</font> being   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font><b> such that </b><a NAME="E9:72"/><i><font color="Green" title="E50">A5</font></i>: 
( <font color="Maroon" title="c4">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c7">Q</font> &amp; <font color="Maroon" title="c5">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c7">Q</font> &amp; <font color="Maroon" title="c6">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c7">Q</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">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="E11:72"/><i><font color="Green" title="E51">A6</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">B</font>
 <b>and </b><br/><a NAME="E12:72"/><i><font color="Green" title="E52">A7</font></i>: 
<font color="Maroon" title="c8">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">L1</font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E50"><i><font color="Green" title="E26">Lm1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">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="E14:72"/><i><font color="Green" title="E53">A8</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">C</font>
 <b>and </b><br/><a NAME="E15:72"/><i><font color="Green" title="E54">A9</font></i>: 
<font color="Maroon" title="c9">C</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L2</font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E50"><i><font color="Green" title="E26">Lm1</font></i></a></i>;<br/>
<a NAME="E16:72"/><i><font color="Green" title="E55">A10</font></i>: 
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c8">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="c5">L1</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c9">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="c6">L2</font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E7:72"><i><font color="Green" title="E49">A4</font></i></a>, <a class="txt" href="incsp_1.html#E12:72"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="incsp_1.html#E15:72"><i><font color="Green" title="E54">A9</font></i></a>, <a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a></i>;<br/>
<a NAME="E17:72"/><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="c2">A</font>,<font color="Maroon" title="c8">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="c3">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="c2">A</font>,<font color="Maroon" title="c8">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="c7">Q</font> &amp; <font color="Maroon" title="c9">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c9">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c7">Q</font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E6:72"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="incsp_1.html#E9:72"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="incsp_1.html#E15:72"><i><font color="Green" title="E54">A9</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self" title="INCSP_1:def.8">Def8</a>, <a class="ref" href="incsp_1.html#T35" target="_self" title="INCSP_1:th.35">Th35</a></i>;<br/>
<a NAME="E18:72"/><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="c2">A</font>,<font color="Maroon" title="c8">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="c9">C</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="c3">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="c2">A</font>,<font color="Maroon" title="c8">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="c9">C</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="c7">Q</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T19" target="_self" title="INCSP_1:th.19">Th19</a></i>;<br/>
<a NAME="E19:72"/><b>then </b><i><font color="Green" title="E56">A11</font></i>: 
( <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c9">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="c3">P</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c9">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="c7">Q</font> )
 
<b>by </b><i><a class="ref" href="enumset1.html#T43" title="ENUMSET1:th.43">ENUMSET1:43</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c10">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="E1:72_1"/><i><font color="Green" title="E57">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="c2">A</font>,<font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c9">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="c10">K</font>
 ;<br/><a NAME="E2:72_1"/>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c8">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="c10">K</font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E1:72_1"><i><font color="Green" title="E57">A12</font></i></a>, <a class="ref" href="enumset1.html#T98" title="ENUMSET1:th.98">ENUMSET1:98</a></i>;<br/><a NAME="E3:72_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c9">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="c8">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="c10">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="c2">A</font>,<font color="Maroon" title="c8">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="c9">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="c10">K</font> )
 <b>by </b><i><a class="txt" href="incsp_1.html#E1:72_1"><i><font color="Green" title="E57">A12</font></i></a>, <a class="ref" href="enumset1.html#T43" title="ENUMSET1:th.43">ENUMSET1:43</a></i>;<br/><a NAME="E4:72_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c8">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="c10">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="c2">A</font>,<font color="Maroon" title="c9">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="c10">K</font> )
 <b>by </b><i><a class="ref" href="incsp_1.html#T18" target="_self" title="INCSP_1:th.18">Th18</a></i>;<br/><a NAME="E5:72_1"/><b>then </b>
( <font color="Maroon" title="c10">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L1</font> &amp; <font color="Maroon" title="c10">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L2</font> )
 <b>by </b><i><a class="txt" href="incsp_1.html#E11:72"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="incsp_1.html#E14:72"><i><font color="Green" title="E53">A8</font></i></a>, <a class="txt" href="incsp_1.html#E16:72"><i><font color="Green" title="E55">A10</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self" title="INCSP_1:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E6:72_1"/>
contradiction
 <b>by </b><i><a class="txt" href="incsp_1.html#E5:72"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:72"/><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="c2">A</font>,<font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c9">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#NR6" title="INCSP_1:NR.6">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="incsp_1.html#D6" target="_self" title="INCSP_1:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E22:72"/>
contradiction
 <b>by </b><i><a class="txt" href="incsp_1.html#E6:72"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="incsp_1.html#E9:72"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="incsp_1.html#E19:72"><i><font color="Green" title="E56">A11</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self" title="INCSP_1:def.8">Def8</a></i>;<br/>


</div>
