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

<b>let </b><font color="Maroon" title="c1">CS</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="collsp.html#L1" title="COLLSP:struct.1">CollStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">CS</font> is  <a href="anproj_2.html#V7" title="ANPROJ_2:attr.7">2-dimensional</a> <a href="anproj_2.html#NM1" title="ANPROJ_2:NM.1">CollProjectiveSpace</a> iff ( <font color="Maroon" title="c1">CS</font> is  <a href="collsp.html#V4" title="COLLSP:attr.4">proper</a> <a href="anproj_2.html#V3" title="ANPROJ_2:attr.3">at_least_3rank</a> <a href="collsp.html#NM2" title="COLLSP:NM.2">CollSp</a> &amp; ( for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">q</font>, <font color="Olive" title="b4">q1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font>  ex <font color="Olive" title="b5">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font> st <br/>( <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b5">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">q</font>,<font color="Olive" title="b4">q1</font>,<font color="Olive" title="b5">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  ) ) ) )</span><br/>

<b>thus </b><a NAME="E1:103"/>
( <font color="Maroon" title="c1">CS</font> is  <a href="anproj_2.html#V7" title="ANPROJ_2:attr.7">2-dimensional</a> <a href="anproj_2.html#NM1" title="ANPROJ_2:NM.1">CollProjectiveSpace</a> implies ( <font color="Maroon" title="c1">CS</font> is  <a href="collsp.html#V4" title="COLLSP:attr.4">proper</a> <a href="anproj_2.html#V3" title="ANPROJ_2:attr.3">at_least_3rank</a> <a href="collsp.html#NM2" title="COLLSP:NM.2">CollSp</a> &amp; ( for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">q</font>, <font color="Olive" title="b4">q1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font>  ex <font color="Olive" title="b5">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font> st <br/>( <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b5">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">q</font>,<font color="Olive" title="b4">q1</font>,<font color="Olive" title="b5">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  ) ) ) )
 <b>by </b><i><a class="ref" href="anproj_2.html#D14" target="_self" title="ANPROJ_2:def.14">Def14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">CS</font> is  <a href="collsp.html#V4" title="COLLSP:attr.4">proper</a> <a href="anproj_2.html#V3" title="ANPROJ_2:attr.3">at_least_3rank</a> <a href="collsp.html#NM2" title="COLLSP:NM.2">CollSp</a> &amp; ( for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">q</font>, <font color="Olive" title="b4">q1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font>  ex <font color="Olive" title="b5">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font> st <br/>( <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b5">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">q</font>,<font color="Olive" title="b4">q1</font>,<font color="Olive" title="b5">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  ) ) implies <font color="Maroon" title="c1">CS</font> is  <a href="anproj_2.html#V7" title="ANPROJ_2:attr.7">2-dimensional</a> <a href="anproj_2.html#NM1" title="ANPROJ_2:NM.1">CollProjectiveSpace</a> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:103"/><i><font color="Green" title="E90">A1</font></i>: 
<font color="Maroon" title="c1">CS</font> is  <a href="collsp.html#V4" title="COLLSP:attr.4">proper</a> <a href="anproj_2.html#V3" title="ANPROJ_2:attr.3">at_least_3rank</a> <a href="collsp.html#NM2" title="COLLSP:NM.2">CollSp</a>
 <b>and </b><br/><a NAME="E3:103"/><i><font color="Green" title="E91">A2</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">q</font>, <font color="Olive" title="b4">q1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font>  ex <font color="Olive" title="b5">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font> st <br/>( <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b5">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">q</font>,<font color="Olive" title="b4">q1</font>,<font color="Olive" title="b5">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">CS</font> is  <a href="anproj_2.html#V7" title="ANPROJ_2:attr.7">2-dimensional</a> <a href="anproj_2.html#NM1" title="ANPROJ_2:NM.1">CollProjectiveSpace</a></span><br/>

<a NAME="E4:103"/>
<font color="Maroon" title="c1">CS</font> is <a href="anproj_2.html#V2" title="ANPROJ_2:attr.2">Vebleian</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font>, <font color="Maroon" title="c5">r</font>, <font color="Maroon" title="c6">r1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="anproj_2.html#D9" target="_self" title="ANPROJ_2:def.9">ANPROJ_2:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  implies  ex <font color="Olive" title="b1">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font> st <br/>( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">r1</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  ) )</span><br/>

<b>assume </b><a NAME="E1:103_1"/>
( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font> st <br/>( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">r1</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/>

<b>thus </b><a NAME="E2:103_1"/>
 ex <font color="Olive" title="b1">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font> st <br/>( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">r1</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:103"><i><font color="Green" title="E91">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:103"/>
<font color="Maroon" title="c1">CS</font> is  <a href="anproj_2.html#V7" title="ANPROJ_2:attr.7">2-dimensional</a> <a href="anproj_2.html#NM1" title="ANPROJ_2:NM.1">CollProjectiveSpace</a>
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:103"><i><font color="Green" title="E90">A1</font></i></a>, <a class="txt" href="anproj_2.html#E3:103"><i><font color="Green" title="E91">A2</font></i></a>, <a class="ref" href="anproj_2.html#D14" target="_self" title="ANPROJ_2:def.14">Def14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
