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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="incsp_1.html#NM4">IncSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="incsp_1.html#NM3">PLANE</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:77"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>given </b><font color="Maroon">c<sub>4</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E2:77"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:77"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="incsp_1.html#E2:77"><i><font color="Green">E47</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>6</sub></font> =  <a href="incsp_1.html#K1" target="_self">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>

<a NAME="E5:77"/>
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E2:77"><i><font color="Green">E47</font></i></a>, <a class="txt" href="incsp_1.html#E4:77"><i><font color="Green">E48</font></i></a>, <a class="ref" href="incsp_1.html#D9" target="_self">Def9</a>, <a class="ref" href="incsp_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E6:77"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E4:77"><i><font color="Green">E48</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>thus </b><a NAME="E7:77"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>6</sub></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">E50</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>and </b><br/><a NAME="E2:77_1"/><i><font color="Green">E51</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="incsp_1.html#NM3">PLANE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:77_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM3">PLANE</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( ( <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Olive">b<sub>1</sub></font> ) iff <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="incsp_1.html#E2:77_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="incsp_1.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E5:77_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E6:77"><i><font color="Green">E49</font></i></a>, <a class="txt" href="incsp_1.html#E1:77_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="incsp_1.html#E4:77_1"><i><font color="Green">E52</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">E46</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E8:77"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>6</sub></font> implies ( <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="incsp_1.html#E6:77"><i><font color="Green">E49</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/>


</div>
