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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>;<br/><b>let </b><font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green" title="E47">A2</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a> 
 ;<br/>

<div><i><font color="Green" title="E48">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_1"/><i><font color="Green" title="E49">A4</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">X</font>
 ;<br/><b>consider </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E3:54_1"/><i><font color="Green" title="E50">A5</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>and </b><br/><a NAME="E4:54_1"/>
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E2:54"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="aff_4.html#T34" target="_self" title="AFF_4:th.34">Th34</a></i>;<br/><b>take </b><font color="Maroon" title="c7">N</font> = <font color="Maroon" title="c4">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c2">M</font>;<br/><b>thus </b><a NAME="E5:54_1"/>
<font color="Maroon" title="c7">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:54"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="aff_4.html#E1:54_1"><i><font color="Green" title="E49">A4</font></i></a>, <a class="txt" href="aff_4.html#E3:54_1"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="aff_4.html#D4" target="_self" title="AFF_4:def.4">Def4</a></i>;<br/><b>thus </b><a NAME="E6:54_1"/>
( <font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">N</font> or <font color="Maroon" title="c7">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:54"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c4">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><a NAME="E1:54_2"/><i><font color="Green" title="E49">A6</font></i>: 
<font color="Maroon" title="c4">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 <b>and </b><br/><a NAME="E2:54_2"/><i><font color="Green" title="E50">A7</font></i>: 
( <font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">N</font> or <font color="Maroon" title="c4">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">M</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>;<br/><b>let </b><font color="Maroon" title="c6">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:54_2_1"/><i><font color="Green" title="E51">A8</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>and </b><br/><a NAME="E2:54_2_1"/><i><font color="Green" title="E52">A9</font></i>: 
( <font color="Maroon" title="c6">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font> )
 ;<br/><a NAME="E3:54_2_1"/><i><font color="Green" title="E53">A10</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c6">A</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:54_2_1"><i><font color="Green" title="E52">A9</font></i></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></i>;<br/><a NAME="E4:54_2_1"/>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:54"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="aff_4.html#E2:54_2_1"><i><font color="Green" title="E52">A9</font></i></a>, <a class="ref" href="aff_4.html#T33" target="_self" title="AFF_4:th.33">Th33</a></i>;<br/><a NAME="E5:54_2_1"/><b>then </b>
<font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c6">A</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:54_2_1"><i><font color="Green" title="E52">A9</font></i></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></i>;<br/><a NAME="E6:54_2_1"/><b>then </b>
<font color="Maroon" title="c4">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c6">A</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:54_2"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/><b>hence </b><a NAME="E7:54_2_1"/>
<font color="Maroon" title="c5">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:54"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="aff_4.html#E1:54_2"><i><font color="Green" title="E49">A6</font></i></a>, <a class="txt" href="aff_4.html#E1:54_2_1"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="aff_4.html#E3:54_2_1"><i><font color="Green" title="E53">A10</font></i></a>, <a class="ref" href="aff_4.html#T23" target="_self" title="AFF_4:th.23">Th23</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:54_2"/>
<font color="Maroon" title="c2">M</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="ref" href="aff_4.html#D4" target="_self" title="AFF_4:def.4">Def4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:54"/>
( <font color="Maroon" title="c2">M</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">X</font> iff  ex <font color="Olive" title="b1">N</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b1">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> &amp; ( <font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b1">N</font> or <font color="Olive" title="b1">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">M</font> ) ) )
 <b>by </b><i><a class="txt" href="aff_4.html#E3:54"><i><font color="Green" title="E48">A3</font></i></a></i>;<br/>


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