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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> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Olive" title="b3">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">A</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">X</font> &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="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Olive" title="b2">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X</font> &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="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c4">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &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="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:55"/><i><font color="Green" title="E46">A1</font></i>: 
( <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="c4">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 <b>and </b><br/><a NAME="E2:55"/><i><font color="Green" title="E47">A2</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )
 <b>and </b><br/><a NAME="E3:55"/><i><font color="Green" title="E48">A3</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="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="E5:55"/><i><font color="Green" title="E49">A4</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="E6:55"/><i><font color="Green" title="E50">A5</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="#E1:55"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="#E3:55"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="aff_4.html#T41" target="_self" title="AFF_4:th.41">Th41</a></i>;<br/>
<a NAME="E7:55"/><i><font color="Green" title="E51">A6</font></i>: 
<font color="Maroon" title="c5">N</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>
 
<b>by </b><i><a class="txt" href="#E6:55"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/>
<font color="Maroon" title="c3">A</font> = 
<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>
<b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="#E2:55"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="aff_4.html#E42"><i><font color="Green" title="E35">Lm8</font></i></a></i>
<br/>.= 
<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="#E6:55"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="aff_4.html#T32" target="_self" title="AFF_4:th.32">Th32</a></i>
;<br/>
<b>hence </b><a NAME="E9:55"/>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="#E2:55"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="#E5:55"><i><font color="Green" title="E49">A4</font></i></a>, <a class="txt" href="#E7:55"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="aff_4.html#T28" target="_self" title="AFF_4:th.28">Th28</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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