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

<span class="kw">let </span><font color="Maroon" title="c1">AS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">a9</font>, <font color="Olive" title="b5">b9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">a9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b5">b9</font> &amp; <font color="Olive" title="b4">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">b9</font> holds <br/>( <font color="Olive" title="b4">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o</font> &amp; <font color="Olive" title="b5">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">a9</font>, <font color="Maroon" title="c6">b9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">a9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">b9</font> &amp; <font color="Maroon" title="c5">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b9</font> implies ( <font color="Maroon" title="c5">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</font> &amp; <font color="Maroon" title="c6">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</font> ) )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:77"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">and </span><br/><a NAME="E2:77"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">a9</font>
 <span class="kw">and </span><br/><a NAME="E3:77"/><span class="lab"><font color="Green" title="E54">A3</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">b9</font>
 <span class="kw">and </span><br/><a NAME="E4:77"/><span class="lab"><font color="Green" title="E55">A4</font></span>: 
<font color="Maroon" title="c5">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</font> &amp; <font color="Maroon" title="c6">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">C</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font>;<br/>
<a NAME="E5:77"/><span class="lab"><font color="Green" title="E56">A5</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" target="_self" title="AFF_1:th.26">Th26</a></span>;<br/>
<a NAME="E6:77"/><span class="lab"><font color="Green" title="E57">A6</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="aff_1.html#T16" target="_self" title="AFF_1:th.16">Th16</a></span>;<br/>
<a NAME="E7:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A7</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D3" target="_self" title="AFF_1:def.3">Def3</a></span>;<br/>
<a NAME="E8:77"/><span class="lab"><font color="Green" title="E59">A8</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="aff_1.html#T16" target="_self" title="AFF_1:th.16">Th16</a></span>;<br/>
<a NAME="E9:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A9</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D3" target="_self" title="AFF_1:def.3">Def3</a></span>;<br/>
<a NAME="E10:77"/><span class="lab"><font color="Green" title="E61">A10</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" target="_self" title="AFF_1:th.26">Th26</a></span>;<br/>
<a NAME="E11:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A11</font></span>: 
<font color="Maroon" title="c5">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E8:77"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E9:77"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E5:77"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="aff_1.html#T39" target="_self" title="AFF_1:th.39">Th39</a></span>;<br/>
<a NAME="E12:77"/><span class="lab"><font color="Green" title="E63">A12</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" target="_self" title="AFF_1:th.26">Th26</a></span>;<br/>
<a NAME="E13:77"/><span class="lab"><font color="Green" title="E64">A13</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" target="_self" title="AFF_1:th.26">Th26</a></span>;<br/>
<a NAME="E14:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A14</font></span>: 
<font color="Maroon" title="c6">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E6:77"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="txt" href="#E7:77"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="txt" href="#E12:77"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="ref" href="aff_1.html#T39" target="_self" title="AFF_1:th.39">Th39</a></span>;<br/>
<a NAME="E15:77"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E9:77"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E5:77"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E10:77"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="txt" href="#E12:77"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="ref" href="aff_1.html#T33" target="_self" title="AFF_1:th.33">Th33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:77"/>
( <font color="Maroon" title="c5">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</font> &amp; <font color="Maroon" title="c6">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E9:77"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E7:77"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="txt" href="#E5:77"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E13:77"><span class="lab"><font color="Green" title="E64">A13</font></span></a>, <a class="txt" href="#E14:77"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E11:77"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="ref" href="aff_1.html#T30" target="_self" title="AFF_1:th.30">Th30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
