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<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="firebrick">::  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="firebrick">::  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="E1:65"/><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="E2:65"/><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="E3:65"/><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="E4:65"/><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="firebrick">::  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="E5:65"/><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#T15" target="_self" title="AFF_1:th.15">Th26</a></span>;<br/>
<a NAME="E6:65"/><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="#E1:65"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="aff_1.html#T7" target="_self" title="AFF_1:th.7">Th16</a></span>;<br/>
<a NAME="E7:65"/><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="E8:65"/><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="#E1:65"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="aff_1.html#T7" target="_self" title="AFF_1:th.7">Th16</a></span>;<br/>
<a NAME="E9:65"/><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="E10:65"/><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#T15" target="_self" title="AFF_1:th.15">Th26</a></span>;<br/>
<a NAME="E11:65"/><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="#E2:65"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E8:65"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E9:65"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E5:65"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="aff_1.html#T25" target="_self" title="AFF_1:th.25">Th39</a></span>;<br/>
<a NAME="E12:65"/><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#T15" target="_self" title="AFF_1:th.15">Th26</a></span>;<br/>
<a NAME="E13:65"/><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#T15" target="_self" title="AFF_1:th.15">Th26</a></span>;<br/>
<a NAME="E14:65"/><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="#E3:65"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E6:65"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="txt" href="#E7:65"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="txt" href="#E12:65"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="ref" href="aff_1.html#T25" target="_self" title="AFF_1:th.25">Th39</a></span>;<br/>
<a NAME="E15:65"/>
 <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="#E1:65"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E9:65"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E5:65"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E10:65"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="txt" href="#E12:65"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="ref" href="aff_1.html#T21" target="_self" title="AFF_1:th.21">Th33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:65"/>
( <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="#E4:65"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E9:65"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E7:65"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="txt" href="#E5:65"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E13:65"><span class="lab"><font color="Green" title="E64">A13</font></span></a>, <a class="txt" href="#E14:65"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E11:65"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="ref" href="aff_1.html#T18" target="_self" title="AFF_1:th.18">Th30</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/>


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