<?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">P</font>, <font color="Maroon" title="c3">Q</font> be   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font>;<br/>





<b>assume </b><a NAME="E1:77"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c2">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">Q</font>
 ;<br/>

<b>given </b><font color="Maroon" title="c4">A</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><b> such that </b><a NAME="E2:77"/><i><font color="Green" title="E47">A2</font></i>: 
( <font color="Maroon" title="c4">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c4">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">Q</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">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="E4:77"/><i><font color="Green" title="E48">A3</font></i>: 
( <font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c5">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c5">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">Q</font> )
 <b>by </b><i><a class="txt" href="incsp_1.html#E2:77"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self" title="INCSP_1:def.8">Def8</a></i>;<br/>
<b>take </b><font color="Maroon" title="c6">L</font> =  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">C</font>;<br/>

<a NAME="E5:77"/>
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">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">L</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">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> <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="c4">A</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c3">Q</font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E2:77"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="incsp_1.html#E4:77"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="incsp_1.html#D9" target="_self" title="INCSP_1:def.9">Def9</a>, <a class="ref" href="incsp_1.html#T13" target="_self" title="INCSP_1:th.13">Th13</a></i>;<br/>
<a NAME="E6:77"/><b>then </b><i><font color="Green" title="E49">A4</font></i>: 
( <font color="Maroon" title="c6">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="c6">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">Q</font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E4:77"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self" title="INCSP_1:def.8">Def8</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">B</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>thus </b><a NAME="E7:77"/>
( <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">Q</font> implies <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:77_1"/><i><font color="Green" title="E50">A5</font></i>: 
( <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">Q</font> )
 <b>and </b><br/><a NAME="E2:77_1"/><i><font color="Green" title="E51">A6</font></i>: 
not <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">P1</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="E4:77_1"/><i><font color="Green" title="E52">A7</font></i>: 
for <font color="Olive" title="b1">P2</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( ( <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b1">P2</font> &amp; <font color="Maroon" title="c6">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">P2</font> ) iff <font color="Maroon" title="c8">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P2</font> )
 <b>by </b><i><a class="txt" href="incsp_1.html#E2:77_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="incsp_1.html#T48" target="_self" title="INCSP_1:th.48">Th48</a></i>;<br/>
<a NAME="E5:77_1"/>
( <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P1</font> &amp; <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P1</font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E6:77"><i><font color="Green" title="E49">A4</font></i></a>, <a class="txt" href="incsp_1.html#E1:77_1"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="incsp_1.html#E4:77_1"><i><font color="Green" title="E52">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:77_1"/>
contradiction
 <b>by </b><i><a class="txt" href="incsp_1.html#E1:77"><i><font color="Green" title="E46">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E8:77"/>
( <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font> implies ( <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">Q</font> ) )
 <b>by </b><i><a class="txt" href="incsp_1.html#E6:77"><i><font color="Green" title="E49">A4</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self" title="INCSP_1:def.8">Def8</a></i>;<br/>


</div>
