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<b>let </b><font color="Maroon">AS</font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/>





<b>assume </b><a NAME="E1:15"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">X</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">Y</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 ;<br/>

<a NAME="E2:15"/><b>then </b><i><font color="Green">E26</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K1" target="_self">AfLines</a> <font color="Maroon">AS</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K1" target="_self">AfLines</a> <font color="Maroon">AS</font> )
 
;<br/>
<a NAME="E3:15"/><i><font color="Green">E30</font></i>: 
(  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K3" target="_self">LinesParallelity</a> <font color="Maroon">AS</font></span>)</span>,<font color="Maroon">X</font> &amp;  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K3" target="_self">LinesParallelity</a> <font color="Maroon">AS</font></span>)</span>,<font color="Maroon">Y</font> )
 
<b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1"/>
 <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E2:15_1"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K3" target="_self">LinesParallelity</a> <font color="Maroon">AS</font></span>)</span>,<font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/><a NAME="E3:15_1"/><b>then </b>
 ex <font color="Olive">Y'</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y'</font> &amp; <font color="Olive">Y'</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">Y</font> <a href="aff_4.html#R1">'||'</a> <font color="Olive">Y'</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:15_1"/>
<font color="Maroon">X</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="aff_4.html#T40">AFF_4:40</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2"/>
<font color="Maroon">X</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E2:15_2"/><b>then </b>
<font color="Maroon">X</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="aff_4.html#T40">AFF_4:40</a></i>;<br/><a NAME="E3:15_2"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K3" target="_self">LinesParallelity</a> <font color="Maroon">AS</font></span>)</span>,<font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:15_2"/>
 <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:15"/>
(  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y</font> iff <font color="Maroon">X</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/>


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