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

<b>consider </b><font color="Maroon" title="c1">V</font> being  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a><b> such that </b><br/><a NAME="E2:101_1_1"/><i><font color="Green" title="E90">A1</font></i>: 
 ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( ( for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font>, <font color="Olive" title="b6">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b2">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">c</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> holds <br/>( <font color="Olive" title="b4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Olive" title="b5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Olive" title="b6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) ) &amp; ( for <font color="Olive" title="b4">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  ex <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b2">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">c</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b3">w</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="anproj_2.html#T16" target="_self" title="ANPROJ_2:th.16">Th16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:101_1_1"/><i><font color="Green" title="E91">A2</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">c</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> holds <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>and </b><br/><a NAME="E5:101_1_1"/>
for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</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="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">c</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:101_1_1"><i><font color="Green" title="E90">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">V</font> = <font color="Maroon" title="c1">V</font> as  <a href="anproj_2.html#V1" title="ANPROJ_2:attr.1">up-3-dimensional</a> <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a> <b>by </b><i><a class="txt" href="anproj_2.html#E4:101_1_1"><i><font color="Green" title="E91">A2</font></i></a>, <a class="ref" href="anproj_2.html#D6" target="_self" title="ANPROJ_2:def.6">Def6</a></i>;<br/>
<b>take </b><font color="Maroon" title="c6">CS</font> =  <a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c5">V</font>;<br/>

<b>thus </b><a NAME="E7:101_1_1"/>
( <font color="Maroon" title="c6">CS</font> is <a href="collsp.html#V1" title="COLLSP:attr.1">strict</a> &amp; <font color="Maroon" title="c6">CS</font> is <a href="anproj_2.html#V4" title="ANPROJ_2:attr.4">Fanoian</a> &amp; <font color="Maroon" title="c6">CS</font> is <a href="anproj_2.html#V5" title="ANPROJ_2:attr.5">Desarguesian</a> &amp; <font color="Maroon" title="c6">CS</font> is <a href="anproj_2.html#V6" title="ANPROJ_2:attr.6">Pappian</a> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">CS1</font> being   <a href="anproj_2.html#NM1" title="ANPROJ_2:NM.1">CollProjectiveSpace</a><b> such that </b><br/><a NAME="E9:101_1_1"/><i><font color="Green" title="E92">A3</font></i>: 
<font color="Maroon" title="c7">CS1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c5">V</font>
 <b>and </b><br/><a NAME="E10:101_1_1"/><i><font color="Green" title="E93">A4</font></i>: 
<font color="Maroon" title="c7">CS1</font> is <a href="anproj_2.html#V7" title="ANPROJ_2:attr.7">2-dimensional</a>
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:101_1_1"><i><font color="Green" title="E90">A1</font></i></a>, <a class="ref" href="anproj_2.html#T31" target="_self" title="ANPROJ_2:th.31">Th31</a></i>;<br/>
<b>thus </b><a NAME="E11:101_1_1"/>
<font color="Maroon" title="c6">CS</font> is <a href="anproj_2.html#V7" title="ANPROJ_2:attr.7">2-dimensional</a>
 <b>by </b><i><a class="txt" href="anproj_2.html#E9:101_1_1"><i><font color="Green" title="E92">A3</font></i></a>, <a class="txt" href="anproj_2.html#E10:101_1_1"><i><font color="Green" title="E93">A4</font></i></a></i>;<br/>


</div>
