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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> is <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a> implies for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">a'</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">b'</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">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">a</font>,<font color="Olive" title="b2">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">b</font> &amp; not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">c</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">b'</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">c'</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">a'</font>,<font color="Olive" title="b4">b'</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">a'</font>,<font color="Olive" title="b6">c'</font> holds <br/><font color="Olive" title="b3">b</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">b'</font>,<font color="Olive" title="b6">c'</font> )</span><br/>

<b>assume </b><a NAME="E1:3"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c1">SAS</font> is <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</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">a</font>, <font color="Olive" title="b2">a'</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">b'</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">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">a</font>,<font color="Olive" title="b2">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">b</font> &amp; not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">c</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">b'</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">c'</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">a'</font>,<font color="Olive" title="b4">b'</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">a'</font>,<font color="Olive" title="b6">c'</font> holds <br/><font color="Olive" title="b3">b</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">b'</font>,<font color="Olive" title="b6">c'</font></span><br/>

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

<b>assume </b><b>that </b><br/><a NAME="E2:3"/><i><font color="Green" title="E5">A2</font></i>: 
( not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font> &amp; not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">c</font> )
 <b>and </b><br/><a NAME="E3:3"/><i><font color="Green" title="E6">A3</font></i>: 
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b'</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c'</font> )
 <b>and </b><br/><a NAME="E4:3"/><i><font color="Green" title="E7">A4</font></i>: 
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c5">b'</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c7">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="c4">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">c'</font></span><br/>

<a NAME="E5:3"/><i><font color="Green" title="E8">A5</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a'</font>
 
<b>by </b><i><a class="txt" href="#E2:3"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>;<br/>
<a NAME="E6:3"/><i><font color="Green" title="E9">A6</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">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">a</font>,<font color="Maroon" title="c3">a'</font> &amp; <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">a</font>,<font color="Maroon" title="c3">a'</font> )
 
<b>by </b><i><a class="txt" href="#E5:3"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/>
<a NAME="E7:3"/><b>then </b><i><font color="Green" title="E10">A7</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">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="E9:3"/><i><font color="Green" title="E11">A8</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">P</font>
 <b>and </b><br/><a NAME="E10:3"/><i><font color="Green" title="E12">A9</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c9">P</font>
 <b>by </b><i><a class="txt" href="#E6:3"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">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="E12:3"/><i><font color="Green" title="E13">A10</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C</font>
 <b>and </b><br/><a NAME="E13:3"/><i><font color="Green" title="E14">A11</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">C</font>
 <b>by </b><i><a class="txt" href="#E6:3"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/>
<a NAME="E14:3"/><i><font color="Green" title="E15">A12</font></i>: 
( <font color="Maroon" title="c9">P</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c10">C</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 
<b>by </b><i><a class="txt" href="#E10:3"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="#E13:3"><i><font color="Green" title="E14">A11</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/>
<a NAME="E15:3"/>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c9">P</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c10">C</font> )
 
<b>by </b><i><a class="txt" href="#E7:3"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="#E10:3"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="#E13:3"><i><font color="Green" title="E14">A11</font></i></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></i>;<br/>
<a NAME="E16:3"/><b>then </b>
( <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c9">P</font> &amp; <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c10">C</font> )
 
<b>by </b><i><a class="txt" href="#E3:3"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="#E5:3"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></i>;<br/>
<a NAME="E17:3"/><b>then </b><i><font color="Green" title="E16">A13</font></i>: 
( <font color="Maroon" title="c5">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">P</font> &amp; <font color="Maroon" title="c7">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C</font> )
 
<b>by </b><i><a class="txt" href="#E9:3"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="#E12:3"><i><font color="Green" title="E13">A10</font></i></a>, <a class="txt" href="#E14:3"><i><font color="Green" title="E15">A12</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/>
<a NAME="E18:3"/><i><font color="Green" title="E17">A14</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">P</font>
 
<b>by </b><i><a class="txt" href="#E2:3"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="#E6:3"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="#E9:3"><i><font color="Green" title="E11">A8</font></i></a>, <a class="ref" href="aff_1.html#T65" title="AFF_1:th.65">AFF_1:65</a></i>;<br/>
<a NAME="E19:3"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">C</font>
 
<b>by </b><i><a class="txt" href="#E2:3"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="#E6:3"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="#E12:3"><i><font color="Green" title="E13">A10</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="E20:3"/>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">c'</font>
 <b>by </b><i><a class="txt" href="#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="#E4:3"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="#E6:3"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="#E9:3"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="#E10:3"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="#E12:3"><i><font color="Green" title="E13">A10</font></i></a>, <a class="txt" href="#E13:3"><i><font color="Green" title="E14">A11</font></i></a>, <a class="txt" href="#E14:3"><i><font color="Green" title="E15">A12</font></i></a>, <a class="txt" href="#E17:3"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="#E18:3"><i><font color="Green" title="E17">A14</font></i></a>, <a class="ref" href="aff_2.html#D11" title="AFF_2:def.11">AFF_2:def 11</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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