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<b>let </b><font color="Maroon" title="c1">SAS</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">SAS</font> <a href="aff_2.html#NR4" title="AFF_2:NR.4">satisfies_DES</a>  implies for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">a'</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">b'</font>, <font color="Olive" title="b6">c</font>, <font color="Olive" title="b7">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font>  st not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b4">b</font> &amp; not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b6">c</font> &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">a'</font> &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b5">b'</font> &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b7">c'</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">a'</font>,<font color="Olive" title="b5">b'</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">a'</font>,<font color="Olive" title="b7">c'</font> holds <br/><font color="Olive" title="b4">b</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b'</font>,<font color="Olive" title="b7">c'</font> )</span><br/>

<b>assume </b><a NAME="E1:2"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c1">SAS</font> <a href="aff_2.html#NR4" title="AFF_2:NR.4">satisfies_DES</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">a'</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">b'</font>, <font color="Olive" title="b6">c</font>, <font color="Olive" title="b7">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font>  st not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b4">b</font> &amp; not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b6">c</font> &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">a'</font> &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b5">b'</font> &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b7">c'</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">a'</font>,<font color="Olive" title="b5">b'</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">a'</font>,<font color="Olive" title="b7">c'</font> holds <br/><font color="Olive" title="b4">b</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b'</font>,<font color="Olive" title="b7">c'</font></span><br/>

<b>let </b><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">a'</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">b'</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b</font> &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a'</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">b'</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">c'</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c6">b'</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c8">c'</font> implies <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c8">c'</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:2"/><i><font color="Green" title="E4">A2</font></i>: 
( not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b</font> &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">c</font> )
 <b>and </b><br/><a NAME="E3:2"/><i><font color="Green" title="E5">A3</font></i>: 
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a'</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">b'</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">c'</font> )
 <b>and </b><br/><a NAME="E4:2"/><i><font color="Green" title="E6">A4</font></i>: 
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c6">b'</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c8">c'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c8">c'</font></span><br/>

<a NAME="E5:2"/><i><font color="Green" title="E7">A5</font></i>: 
(  <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">a'</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="c5">b</font>,<font color="Maroon" title="c6">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="c7">c</font>,<font color="Maroon" title="c8">c'</font> )
 
<b>by </b><i><a class="txt" href="pardepap.html#E3:2"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">SAS</font><b> such that </b><br/><a NAME="E7:2"/><i><font color="Green" title="E8">A6</font></i>: 
<font color="Maroon" title="c9">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 <b>and </b><br/><a NAME="E8:2"/><i><font color="Green" title="E9">A7</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c4">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">SAS</font><b> such that </b><br/><a NAME="E10:2"/><i><font color="Green" title="E10">A8</font></i>: 
<font color="Maroon" title="c10">P</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 <b>and </b><br/><a NAME="E11:2"/><i><font color="Green" title="E11">A9</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font> &amp; <font color="Maroon" title="c6">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font> )
 <b>by </b><i><a class="txt" href="pardepap.html#E5:2"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">SAS</font><b> such that </b><br/><a NAME="E13:2"/><i><font color="Green" title="E12">A10</font></i>: 
<font color="Maroon" title="c11">C</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 <b>and </b><br/><a NAME="E14:2"/><i><font color="Green" title="E13">A11</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C</font> &amp; <font color="Maroon" title="c8">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C</font> )
 <b>by </b><i><a class="txt" href="pardepap.html#E5:2"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E15:2"/><i><font color="Green" title="E14">A12</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="c3">a</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">b</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="c7">c</font> )
 
<b>by </b><i><a class="txt" href="pardepap.html#E2:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</a></i>;<br/>
<a NAME="E16:2"/><i><font color="Green" title="E15">A13</font></i>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">P</font>
 
<b>by </b><i><a class="txt" href="pardepap.html#E2:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="pardepap.html#E7:2"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="pardepap.html#E8:2"><i><font color="Green" title="E9">A7</font></i></a>, <a class="txt" href="pardepap.html#E11:2"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="aff_1.html#T65" title="AFF_1:th.65">AFF_1:65</a></i>;<br/>
<a NAME="E17:2"/>
<font color="Maroon" title="c9">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">C</font>
 
<b>by </b><i><a class="txt" href="pardepap.html#E2:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="pardepap.html#E7:2"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="pardepap.html#E8:2"><i><font color="Green" title="E9">A7</font></i></a>, <a class="txt" href="pardepap.html#E14:2"><i><font color="Green" title="E13">A11</font></i></a>, <a class="ref" href="aff_1.html#T65" title="AFF_1:th.65">AFF_1:65</a></i>;<br/>
<b>hence </b><a NAME="E18:2"/>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c8">c'</font>
 <b>by </b><i><a class="txt" href="pardepap.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="pardepap.html#E4:2"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="pardepap.html#E7:2"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="pardepap.html#E8:2"><i><font color="Green" title="E9">A7</font></i></a>, <a class="txt" href="pardepap.html#E10:2"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="pardepap.html#E11:2"><i><font color="Green" title="E11">A9</font></i></a>, <a class="txt" href="pardepap.html#E13:2"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="pardepap.html#E14:2"><i><font color="Green" title="E13">A11</font></i></a>, <a class="txt" href="pardepap.html#E15:2"><i><font color="Green" title="E14">A12</font></i></a>, <a class="txt" href="pardepap.html#E16:2"><i><font color="Green" title="E15">A13</font></i></a>, <a class="ref" href="aff_2.html#D4" title="AFF_2:def.4">AFF_2:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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