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

<b>let </b><font color="Maroon" title="c1">CPS</font> be   <a href="anproj_2.html#NM1" title="ANPROJ_2:NM.1">CollProjectiveSpace</a>;<br/>

<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E24">A1</font></i>: 
for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">p3</font>, <font color="Olive" title="b5">q1</font>, <font color="Olive" title="b6">q2</font>, <font color="Olive" title="b7">q3</font>, <font color="Olive" title="b8">r1</font>, <font color="Olive" title="b9">r2</font>, <font color="Olive" title="b10">r3</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CPS</font>  st <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">p2</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">p3</font> &amp; <font color="Olive" title="b3">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">p3</font> &amp; <font color="Olive" title="b2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">p2</font> &amp; <font color="Olive" title="b2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">p3</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">q2</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">q3</font> &amp; <font color="Olive" title="b6">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">q3</font> &amp; <font color="Olive" title="b5">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">q2</font> &amp; <font color="Olive" title="b5">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">q3</font> &amp; not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b5">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b4">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b5">q1</font>,<font color="Olive" title="b6">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b5">q1</font>,<font color="Olive" title="b7">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b6">q2</font>,<font color="Olive" title="b10">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b5">q1</font>,<font color="Olive" title="b3">p2</font>,<font color="Olive" title="b10">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b7">q3</font>,<font color="Olive" title="b9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b4">p3</font>,<font color="Olive" title="b5">q1</font>,<font color="Olive" title="b9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">p2</font>,<font color="Olive" title="b7">q3</font>,<font color="Olive" title="b8">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b4">p3</font>,<font color="Olive" title="b6">q2</font>,<font color="Olive" title="b8">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/><font color="Olive" title="b8">r1</font>,<font color="Olive" title="b9">r2</font>,<font color="Olive" title="b10">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c4">a2</font>, <font color="Maroon" title="c5">a3</font>, <font color="Maroon" title="c6">b1</font>, <font color="Maroon" title="c7">b2</font>, <font color="Maroon" title="c8">b3</font>, <font color="Maroon" title="c9">c1</font>, <font color="Maroon" title="c10">c2</font>, <font color="Maroon" title="c11">c3</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c12">A1</font>, <font color="Maroon" title="c13">A2</font>, <font color="Maroon" title="c14">A3</font>, <font color="Maroon" title="c15">B1</font>, <font color="Maroon" title="c16">B2</font>, <font color="Maroon" title="c17">B3</font>, <font color="Maroon" title="c18">C1</font>, <font color="Maroon" title="c19">C2</font>, <font color="Maroon" title="c20">C3</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:24"/><i><font color="Green" title="E25">A2</font></i>: 
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</font> <a href="incproj.html#R2" title="INCPROJ:pred.2">are_mutually_different</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c8">b3</font> <a href="incproj.html#R2" title="INCPROJ:pred.2">are_mutually_different</a>  )
 <b>and </b><br/><a NAME="E3:24"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c14">A3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">B3</font>
 <b>and </b><br/><a NAME="E4:24"/><i><font color="Green" title="E27">A4</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">A3</font> &amp; <font color="Maroon" title="c2">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">B3</font> )
 <b>and </b><br/><a NAME="E5:24"/><i><font color="Green" title="E28">A5</font></i>: 
( <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">c1</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c12">A1</font> &amp; <font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c10">c2</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c15">B1</font> &amp; <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c11">c3</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c18">C1</font> &amp; <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c10">c2</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c13">A2</font> &amp; <font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c9">c1</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c16">B2</font> &amp; <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c11">c3</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c19">C2</font> &amp; <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c8">b3</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c14">A3</font> &amp; <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c17">B3</font> &amp; <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c2</font> <a href="incproj.html#R3" title="INCPROJ:pred.3">on</a> <font color="Maroon" title="c20">C3</font> )
 ;<br/>

<a NAME="E6:24"/><i><font color="Green" title="E29">A6</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b3</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a3</font> &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a2</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b2</font> )
 
<b>by </b><i><a class="txt" href="incproj.html#E2:24"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="incproj.html#D6" target="_self" title="INCPROJ:def.6">Def6</a></i>;<br/>
<a NAME="E7:24"/><i><font color="Green" title="E30">A7</font></i>: 
( <font color="Maroon" title="c4">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">A1</font> &amp; <font color="Maroon" title="c8">b3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">A1</font> &amp; <font color="Maroon" title="c9">c1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">A1</font> &amp; <font color="Maroon" title="c5">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">B1</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">B1</font> &amp; <font color="Maroon" title="c10">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">B1</font> &amp; <font color="Maroon" title="c3">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">C1</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">C1</font> &amp; <font color="Maroon" title="c11">c3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">C1</font> &amp; <font color="Maroon" title="c3">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">A2</font> &amp; <font color="Maroon" title="c8">b3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">A2</font> &amp; <font color="Maroon" title="c10">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">A2</font> &amp; <font color="Maroon" title="c5">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">B2</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">B2</font> &amp; <font color="Maroon" title="c9">c1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">B2</font> &amp; <font color="Maroon" title="c4">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">C2</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">C2</font> &amp; <font color="Maroon" title="c11">c3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">C2</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">A3</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">A3</font> &amp; <font color="Maroon" title="c8">b3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">A3</font> &amp; <font color="Maroon" title="c3">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">B3</font> &amp; <font color="Maroon" title="c4">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">B3</font> &amp; <font color="Maroon" title="c5">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">B3</font> &amp; <font color="Maroon" title="c9">c1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">C3</font> &amp; <font color="Maroon" title="c10">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">C3</font> )
 
<b>by </b><i><a class="txt" href="incproj.html#E5:24"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="incproj.html#D7" target="_self" title="INCPROJ:def.7">Def7</a>, <a class="ref" href="incproj.html#D8" target="_self" title="INCPROJ:def.8">Def8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">o'</font> = <font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c22">a1'</font> = <font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c23">a2'</font> = <font color="Maroon" title="c4">a2</font>, <font color="Maroon" title="c24">a3'</font> = <font color="Maroon" title="c5">a3</font>, <font color="Maroon" title="c25">b1'</font> = <font color="Maroon" title="c6">b1</font>, <font color="Maroon" title="c26">b2'</font> = <font color="Maroon" title="c7">b2</font>, <font color="Maroon" title="c27">b3'</font> = <font color="Maroon" title="c8">b3</font>, <font color="Maroon" title="c28">c1'</font> = <font color="Maroon" title="c9">c1</font>, <font color="Maroon" title="c29">c2'</font> = <font color="Maroon" title="c10">c2</font>, <font color="Maroon" title="c30">c3'</font> = <font color="Maroon" title="c11">c3</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CPS</font> ;<br/>
<a NAME="E9:24"/><i><font color="Green" title="E31">A8</font></i>: 
<font color="Maroon" title="c9">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1"/><i><font color="Green" title="E32">A9</font></i>: 
not <font color="Maroon" title="c9">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c2</font>
 ;<br/>

<a NAME="E2:24_1"/><i><font color="Green" title="E33">A10</font></i>: 
not <font color="Maroon" title="c5">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">A3</font>
 
<b>by </b><i><a class="txt" href="incproj.html#E3:24"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="incproj.html#E4:24"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="incproj.html#E6:24"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="incproj.html#E7:24"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></i>;<br/>
<a NAME="E3:24_1"/>
not <font color="Maroon" title="c8">b3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">B3</font>
 
<b>by </b><i><a class="txt" href="incproj.html#E3:24"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="incproj.html#E4:24"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="incproj.html#E6:24"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="incproj.html#E7:24"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></i>;<br/>
<a NAME="E4:24_1"/><b>then </b><i><font color="Green" title="E34">A11</font></i>: 
<font color="Maroon" title="c12">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">A2</font>
 
<b>by </b><i><a class="txt" href="incproj.html#E6:24"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="incproj.html#E7:24"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></i>;<br/>
<a NAME="E5:24_1"/><i><font color="Green" title="E35">A12</font></i>: 
<font color="Maroon" title="c15">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">B2</font>
 
<b>by </b><i><a class="txt" href="incproj.html#E6:24"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="incproj.html#E7:24"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="incproj.html#E2:24_1"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></i>;<br/>
<a NAME="E6:24_1"/><i><font color="Green" title="E36">A13</font></i>: 
<font color="Maroon" title="c9">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b3</font>
 
<b>by </b><i><a class="txt" href="incproj.html#E7:24"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="incproj.html#E1:24_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="incproj.html#E4:24_1"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></i>;<br/>
<a NAME="E7:24_1"/>
<font color="Maroon" title="c9">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a3</font>
 
<b>by </b><i><a class="txt" href="incproj.html#E7:24"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="incproj.html#E1:24_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="incproj.html#E5:24_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E8:24_1"/>
contradiction
 <b>by </b><i><a class="txt" href="incproj.html#E3:24"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="incproj.html#E4:24"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="incproj.html#E6:24"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="incproj.html#E7:24"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="incproj.html#E6:24_1"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:24"/><i><font color="Green" title="E32">A14</font></i>: 
( <font color="Maroon" title="c21">o'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">a2'</font> &amp; <font color="Maroon" title="c21">o'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">a3'</font> &amp; <font color="Maroon" title="c23">a2'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">a3'</font> &amp; <font color="Maroon" title="c22">a1'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">a2'</font> &amp; <font color="Maroon" title="c22">a1'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">a3'</font> &amp; <font color="Maroon" title="c21">o'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c26">b2'</font> &amp; <font color="Maroon" title="c21">o'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c27">b3'</font> &amp; <font color="Maroon" title="c26">b2'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c27">b3'</font> &amp; <font color="Maroon" title="c25">b1'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c26">b2'</font> &amp; <font color="Maroon" title="c25">b1'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c27">b3'</font> )
 
<b>by </b><i><a class="txt" href="incproj.html#E2:24"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="incproj.html#D6" target="_self" title="INCPROJ:def.6">Def6</a></i>;<br/>
<a NAME="E11:24"/><i><font color="Green" title="E33">A15</font></i>: 
not <font color="Maroon" title="c21">o'</font>,<font color="Maroon" title="c22">a1'</font>,<font color="Maroon" title="c25">b1'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_2"/>
<font color="Maroon" title="c21">o'</font>,<font color="Maroon" title="c22">a1'</font>,<font color="Maroon" title="c25">b1'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c31">K</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span><b> such that </b><br/><a NAME="E3:24_2"/><i><font color="Green" title="E34">A16</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c31">K</font> &amp; <font color="Maroon" title="c3">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c31">K</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c31">K</font> )
 <b>by </b><i><a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></i>;<br/>
<a NAME="E4:24_2"/>
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b1</font> )
 
<b>by </b><i><a class="txt" href="incproj.html#E2:24"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="incproj.html#D6" target="_self" title="INCPROJ:def.6">Def6</a></i>;<br/>
<a NAME="E5:24_2"/><b>then </b>
( <font color="Maroon" title="c31">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">A3</font> &amp; <font color="Maroon" title="c31">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">B3</font> )
 
<b>by </b><i><a class="txt" href="incproj.html#E4:24"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="incproj.html#E7:24"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="incproj.html#E3:24_2"><i><font color="Green" title="E34">A16</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E6:24_2"/>
contradiction
 <b>by </b><i><a class="txt" href="incproj.html#E3:24"><i><font color="Green" title="E26">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:24"/>
( <font color="Maroon" title="c21">o'</font>,<font color="Maroon" title="c22">a1'</font>,<font color="Maroon" title="c23">a2'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c21">o'</font>,<font color="Maroon" title="c22">a1'</font>,<font color="Maroon" title="c24">a3'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c21">o'</font>,<font color="Maroon" title="c25">b1'</font>,<font color="Maroon" title="c26">b2'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c21">o'</font>,<font color="Maroon" title="c25">b1'</font>,<font color="Maroon" title="c27">b3'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c22">a1'</font>,<font color="Maroon" title="c26">b2'</font>,<font color="Maroon" title="c30">c3'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c25">b1'</font>,<font color="Maroon" title="c23">a2'</font>,<font color="Maroon" title="c30">c3'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c22">a1'</font>,<font color="Maroon" title="c27">b3'</font>,<font color="Maroon" title="c29">c2'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c24">a3'</font>,<font color="Maroon" title="c25">b1'</font>,<font color="Maroon" title="c29">c2'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c23">a2'</font>,<font color="Maroon" title="c27">b3'</font>,<font color="Maroon" title="c28">c1'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c24">a3'</font>,<font color="Maroon" title="c26">b2'</font>,<font color="Maroon" title="c28">c1'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="incproj.html#E4:24"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="incproj.html#E7:24"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></i>;<br/>
<a NAME="E13:24"/><b>then </b>
<font color="Maroon" title="c28">c1'</font>,<font color="Maroon" title="c29">c2'</font>,<font color="Maroon" title="c30">c3'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="incproj.html#E1:24"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="incproj.html#E10:24"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="incproj.html#E11:24"><i><font color="Green" title="E33">A15</font></i></a></i>;<br/>
<a NAME="E14:24"/><b>then </b>
 ex <font color="Olive" title="b1">K</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span> st <br/>( <font color="Maroon" title="c9">c1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c10">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c11">c3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> )
 
<b>by </b><i><a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></i>;<br/>
<b>hence </b><a NAME="E15:24"/>
<font color="Maroon" title="c11">c3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">C3</font>
 <b>by </b><i><a class="txt" href="incproj.html#E7:24"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="incproj.html#E9:24"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></i>;<br/>


</div>
