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<b>let </b><font color="Maroon">AS</font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/>

<b>assume </b><a NAME="E1:22"/>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AS</font> <a href="subset_1.html#R1">misses</a>  <a href="afproj.html#K7" target="_self">Dir_of_Lines</a> <font color="Maroon">AS</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AS</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K7" target="_self">Dir_of_Lines</a> <font color="Maroon">AS</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E5:22"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">A</font> = <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">X</font>;<br/>
<a NAME="E7:22"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">X</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">X</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="ref" href="aff_4.html#T27">AFF_4:27</a>, <a class="ref" href="aff_4.html#D3">AFF_4:def 3</a></i>;<br/>
<a NAME="E8:22"/><b>then </b>
<font color="Maroon">X</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="aff_4.html#T40">AFF_4:40</a></i>;<br/>
<a NAME="E9:22"/><b>then </b>
<font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E10:22"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_4.html#D3">AFF_4:def 3</a></i>;<br/>


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