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

<b>set </b><font color="Maroon">XX</font> =  <a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font>;<br/>
<a NAME="E1:66_1_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span><br/> for <font color="Olive">A</font>, <font color="Olive">K</font> being  <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span>  st <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">A</font> &amp; <font color="Olive">b</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">A</font> &amp; <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">K</font> &amp; <font color="Olive">b</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">K</font> &amp; not <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> holds <br/><font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font>
 
<b>by </b><i><a class="ref" href="afproj.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E2:66_1_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span>  ex <font color="Olive">A</font> being  <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span> st <br/>( <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">A</font> &amp; <font color="Olive">b</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">A</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:66_1_1"/><i><font color="Green">E30</font></i>: 
not for <font color="Olive">a</font> being  <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span><br/> for <font color="Olive">A</font> being  <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span> holds  <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:66_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">A</font> being  <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span>  ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span> st <br/>( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Olive">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">A</font> &amp; <font color="Olive">b</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">A</font> &amp; <font color="Olive">c</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span>, <font color="Maroon">b</font> being   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span>, <font color="Maroon">c</font> being   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span><b> such that </b><br/><a NAME="E2:66_1_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="afproj.html#T6" target="_self">Th6</a></i>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

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

<b>take </b>
<font color="Maroon">c</font>
;<br/>

<b>thus </b><a NAME="E3:66_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="afproj.html#D6" target="_self">Def6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:66_1_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font>, <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span><br/> for <font color="Olive">M</font>, <font color="Olive">N</font>, <font color="Olive">P</font>, <font color="Olive">Q</font> being  <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span>  st <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">M</font> &amp; <font color="Olive">b</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">M</font> &amp; <font color="Olive">c</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">N</font> &amp; <font color="Olive">d</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">N</font> &amp; <font color="Olive">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">M</font> &amp; <font color="Olive">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">N</font> &amp; <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> &amp; <font color="Olive">c</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> &amp; <font color="Olive">b</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">Q</font> &amp; <font color="Olive">d</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">Q</font> &amp; not <font color="Olive">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> &amp; not <font color="Olive">p</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">Q</font> &amp; <font color="Olive">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">N</font> holds <br/> ex <font color="Olive">q</font> being  <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span> st <br/>( <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> &amp; <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">Q</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:66_1_1"/>
(  <a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font> is <a href="incproj.html#V1">partial</a> &amp;  <a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font> is <a href="incproj.html#V2">linear</a> &amp;  <a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font> is <a href="incproj.html#V3">up-2-dimensional</a> &amp;  <a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font> is <a href="incproj.html#V4">up-3-rank</a> &amp;  <a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font> is <a href="incproj.html#V5">Vebleian</a> )
 <b>by </b><i>, , , <a class="ref" href="afproj.html#D5" target="_self">Def5</a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a>, <a class="ref" href="incproj.html#D10">INCPROJ:def 10</a>, <a class="ref" href="incproj.html#D11">INCPROJ:def 11</a>, <a class="ref" href="incproj.html#D12">INCPROJ:def 12</a>, <a class="ref" href="incproj.html#D13">INCPROJ:def 13</a></i>;<br/>


</div>
