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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="E1:75"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Maroon">Y</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Maroon">X</font> <a href="aff_4.html#R1" target="_self">'||'</a> <font color="Maroon">Y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>, <font color="Maroon">P</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>, <font color="Maroon">M</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>, <font color="Maroon">N</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E3:75"/><i><font color="Green">E35</font></i>: 
( not <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font> or <font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A</font> ) &amp; ( <font color="Maroon">P</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font> or <font color="Maroon">N</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E4:75"/>
not <font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:75_1"/>
<font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:75_1"/><b>then </b>
<font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E3:75_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:75"/>
<font color="Maroon">Y</font> <a href="aff_4.html#R1" target="_self">'||'</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/>


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