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

<b>let </b><font color="Maroon" title="c1">SAS</font> be   <a href="semi_af1.html#NM1" title="SEMI_AF1:NM.1">Semi_Affine_Space</a>;<br/><b>let </b><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">d</font>, <font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font>;<br/>











<a NAME="E1:93"/><i><font color="Green" title="E75">A1</font></i>: 
( <font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  implies <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">d</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:93_1"/><i><font color="Green" title="E76">A2</font></i>: 
<font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 ;<br/>

<div><i><font color="Green" title="E77">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:93_1_1"/>
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">o</font> or <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> )
 ;<br/><a NAME="E2:93_1_1"/><b>then </b>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T113" target="_self" title="SEMI_AF1:th.113">Th113</a></i>;<br/><b>hence </b><a NAME="E3:93_1_1"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a>, <a class="ref" href="semi_af1.html#T14" target="_self" title="SEMI_AF1:th.14">Th14</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:93_1_2"/>
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">o</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> )
 ;<br/><a NAME="E2:93_1_2"/><b>then </b>
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">o</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> &amp; <font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> &amp; <font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:93_1"><i><font color="Green" title="E76">A2</font></i></a>, <a class="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</a>, <a class="ref" href="semi_af1.html#T114" target="_self" title="SEMI_AF1:th.114">Th114</a></i>;<br/><a NAME="E3:93_1_2"/><b>then </b>
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> &amp; <font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">d</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E4:93_1_2"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:93_1"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E2:93_1"><i><font color="Green" title="E77">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:93"/>
( <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">d</font> implies <font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:93_2"/><i><font color="Green" title="E76">A4</font></i>: 
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">d</font>
 ;<br/>

<div><i><font color="Green" title="E77">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:93_2_1"/>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> )
 ;<br/><a NAME="E2:93_2_1"/><b>then </b>
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">o</font> or <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T113" target="_self" title="SEMI_AF1:th.113">Th113</a></i>;<br/><a NAME="E3:93_2_1"/><b>then </b>
<font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a>, <a class="ref" href="semi_af1.html#T14" target="_self" title="SEMI_AF1:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E4:93_2_1"/>
<font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:93_2_2"/>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d</font> )
 ;<br/><a NAME="E2:93_2_2"/><b>then </b>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">d</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T114" target="_self" title="SEMI_AF1:th.114">Th114</a></i>;<br/><a NAME="E3:93_2_2"/><b>then </b>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">o</font> &amp; <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">d</font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:93_2"><i><font color="Green" title="E76">A4</font></i></a>, <a class="ref" href="semi_af1.html#T17" target="_self" title="SEMI_AF1:th.17">Th17</a></i>;<br/><a NAME="E4:93_2_2"/><b>then </b>
( <font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></i>;<br/><a NAME="E5:93_2_2"/><b>then </b>
<font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T20" target="_self" title="SEMI_AF1:th.20">Th20</a></i>;<br/><b>hence </b><a NAME="E6:93_2_2"/>
<font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:93_2"/>
<font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="semi_af1.html#E2:93_2"><i><font color="Green" title="E77">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:93"/>
( <font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">o</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  iff <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">d</font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:93"><i><font color="Green" title="E75">A1</font></i></a></i>;<br/>


</div>
