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

<span class="kw">let </span><font color="Maroon" title="c1">SAS</font> be   <a href="semi_af1.html#NM1" title="SEMI_AF1:NM.1">Semi_Affine_Space</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">d</font>, <font color="Olive" title="b5">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font> holds <br/> ( <font color="Olive" title="b1">o</font>, <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">o</font>), <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> (<font color="Olive" title="b4">d</font>,<font color="Olive" title="b5">c</font>,<font color="Olive" title="b1">o</font>) <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  iff <font color="Olive" title="b3">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">c</font>,<font color="Olive" title="b4">d</font> )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> )</span><br/>











<a NAME="E1:93"/><span class="lab"><font color="Green" title="E74">A1</font></span>: 
( <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_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:93_1"/><span class="lab"><font color="Green" title="E75">A2</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> </span><br/>

<div><span class="lab"><font color="Green" title="E76">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:93_1_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="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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T88" target="_self" title="SEMI_AF1:th.88">Th114</a></span>;<br/><a NAME="E2:93_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A4</font></span>: 
<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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T6" target="_self" title="SEMI_AF1:th.6">Th17</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:93_1_1"/><span class="lab"><font color="Green" title="E78">A5</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:93_1_1"/><span class="lab"><font color="Green" title="E79">A6</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> </span><br/><a NAME="E5:93_1_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="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T88" target="_self" title="SEMI_AF1:th.88">Th114</a></span>;<br/><a NAME="E6:93_1_1"/><span class="kw">then </span>
<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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T6" target="_self" title="SEMI_AF1:th.6">Th17</a></span>;<br/><a NAME="E7:93_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1"><span class="lab"><font color="Green" title="E75">A2</font></span></a>, <a class="txt" href="#E3:93_1_1"><span class="lab"><font color="Green" title="E78">A5</font></span></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></span>;<br/><a NAME="E8:93_1_1"/><span class="kw">then </span>
<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:93_1_1"><span class="lab"><font color="Green" title="E79">A6</font></span></a>, <a class="txt" href="#E2:93_1_1"><span class="lab"><font color="Green" title="E77">A4</font></span></a>, <a class="ref" href="semi_af1.html#T8" target="_self" title="SEMI_AF1:th.8">Th20</a></span>;<br/><span class="kw">hence </span><a NAME="E9:93_1_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:93_1_2"/>
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> </span><br/><a NAME="E2:93_1_2"/><span class="kw">then </span>
( <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>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T87" target="_self" title="SEMI_AF1:th.87">Th113</a></span>;<br/><a NAME="E3:93_1_2"/><span class="kw">then </span>
<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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a>, <a class="ref" href="semi_af1.html#T3" target="_self" title="SEMI_AF1:th.3">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E4:93_1_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:93_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93_1"><span class="lab"><font color="Green" title="E76">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:93"/>
( <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>  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_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:93_2"/><span class="lab"><font color="Green" title="E75">A7</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<div><span class="lab"><font color="Green" title="E76">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:93_2_1"/><span class="lab"><font color="Green" title="E77">A9</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T88" target="_self" title="SEMI_AF1:th.88">Th114</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:93_2_1"/><span class="lab"><font color="Green" title="E78">A10</font></span>: 
<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>)
 <span class="kw">and </span><br/><a NAME="E3:93_2_1"/><span class="lab"><font color="Green" title="E79">A11</font></span>: 
<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>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_2"><span class="lab"><font color="Green" title="E75">A7</font></span></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#T88" target="_self" title="SEMI_AF1:th.88">Th114</a></span>;<br/><a NAME="E5:93_2_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93_2_1"><span class="lab"><font color="Green" title="E78">A10</font></span></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:93_2_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93_2_1"><span class="lab"><font color="Green" title="E79">A11</font></span></a>, <a class="txt" href="#E1:93_2_1"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:93_2_2"/>
( <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>) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><a NAME="E2:93_2_2"/><span class="kw">then </span>
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T87" target="_self" title="SEMI_AF1:th.87">Th113</a></span>;<br/><span class="kw">hence </span><a NAME="E3:93_2_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a>, <a class="ref" href="semi_af1.html#T3" target="_self" title="SEMI_AF1:th.3">Th14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:93_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93_2"><span class="lab"><font color="Green" title="E76">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93"><span class="lab"><font color="Green" title="E74">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
