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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">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">K</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<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><b>that </b><br/><a NAME="E1:5"/><i><font color="Green">E25</font></i>: 
not <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E2:5"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">A</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font> &amp; <font color="Maroon">A</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">K</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font> &amp; <font color="Maroon">K</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">Y</font> )
 <b>and </b><br/><a NAME="E3:5"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <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>  )
 ;<br/>

<a NAME="E4:5"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, <a class="ref" href="aff_4.html#T59">AFF_4:59</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font>, <font color="Maroon">b</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:5"/>
( <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> <a href="aff_4.html#K3">+</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> <a href="aff_4.html#K3">+</a> <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> <a href="aff_4.html#K3">+</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">b</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> <a href="aff_4.html#K3">+</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_4.html#T68">AFF_4:68</a></i>;<br/>
<a NAME="E9:5"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">b</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_4.html#T66">AFF_4:66</a></i>;<br/>
<a NAME="E10:5"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">b</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">A</font> &amp; <font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">K</font> &amp; <font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">b</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">K</font> )
 
<b>by </b><i>, <a class="ref" href="aff_4.html#D3">AFF_4:def 3</a></i>;<br/>
<a NAME="E11:5"/><i><font color="Green">E35</font></i>: 
not <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">K</font>
 ;<br/>

<a NAME="E2:5_1"/><b>then </b>
<font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E3:5_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>
<a NAME="E12:5"/>
( <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">b</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">b</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">K</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E13:5"/>
<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#T55">AFF_4:55</a></i>;<br/>


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