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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><br/> for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b2">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b3">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  &amp; <font color="Olive" title="b2">A</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b3">X</font> holds <br/><font color="Olive" title="b1">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Olive" title="b3">X</font></span><br/><b>let </b><font color="Maroon" title="c2">a</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b1">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b2">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  &amp; <font color="Olive" title="b1">A</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">X</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Olive" title="b2">X</font></span><br/><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c4">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  &amp; <font color="Maroon" title="c3">A</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">X</font> implies <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c4">X</font> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:86"/><i><font color="Green" title="E68">A1</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c4">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  )
 <b>and </b><br/><a NAME="E2:86"/><i><font color="Green" title="E69">A2</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c4">X</font></span><br/>

<b>consider </b><font color="Maroon" title="c5">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E4:86"/><i><font color="Green" title="E70">A3</font></i>: 
<font color="Maroon" title="c5">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 <b>and </b><br/><a NAME="E5:86"/><i><font color="Green" title="E71">A4</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">N</font> or <font color="Maroon" title="c5">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:86"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="aff_4.html#E2:86"><i><font color="Green" title="E69">A2</font></i></a>, <a class="ref" href="aff_4.html#T41" target="_self" title="AFF_4:th.41">Th41</a></i>;<br/>
<a NAME="E6:86"/><i><font color="Green" title="E72">A5</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c5">N</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E5:86"><i><font color="Green" title="E71">A4</font></i></a>, <a class="ref" href="aff_4.html#T32" target="_self" title="AFF_4:th.32">Th32</a></i>;<br/>
<a NAME="E7:86"/><i><font color="Green" title="E73">A6</font></i>: 
<font color="Maroon" title="c5">N</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 
<b>by </b><i><a class="txt" href="aff_4.html#E5:86"><i><font color="Green" title="E71">A4</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/>
<a NAME="E8:86"/>
( <font color="Maroon" title="c4">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c4">X</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:86"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="aff_4.html#D6" target="_self" title="AFF_4:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E9:86"/>
<font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:86"><i><font color="Green" title="E70">A3</font></i></a>, <a class="txt" href="aff_4.html#E6:86"><i><font color="Green" title="E72">A5</font></i></a>, <a class="txt" href="aff_4.html#E7:86"><i><font color="Green" title="E73">A6</font></i></a>, <a class="ref" href="aff_4.html#D4" target="_self" title="AFF_4:def.4">Def4</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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