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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#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</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>let </b><font color="Maroon">Z</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/>















<b>assume </b><a NAME="E1:81"/><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">Z</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> &amp; <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> )
 ;<br/>

<a NAME="E2:81"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">A</font> = <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font>;<br/>
<b>set </b><font color="Maroon">C</font> = <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font>;<br/>
<a NAME="E3:81"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E24"><i><font color="Green">Lemma67</font></i></a>, <a class="txt" href="aff_4.html#E25"><i><font color="Green">Lemma68</font></i></a>, <a class="ref" href="aff_4.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E4:81"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E24"><i><font color="Green">Lemma67</font></i></a>, <a class="txt" href="aff_4.html#E25"><i><font color="Green">Lemma68</font></i></a>, <a class="ref" href="aff_4.html#T18" target="_self">Th18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> )
 ;<br/><a NAME="E2:81_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i><a class="txt" href="aff_4.html#E24"><i><font color="Green">Lemma67</font></i></a>, <a class="txt" href="aff_4.html#E25"><i><font color="Green">Lemma68</font></i></a>, , , </i>;<br/><b>set </b><font color="Maroon">K</font> = <font color="Maroon">c</font> <a href="aff_4.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span>;<br/><a NAME="E3:81_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c</font> <a href="aff_4.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c</font> <a href="aff_4.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/><a NAME="E4:81_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E5:81_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E6:81_1"/>
( <font color="Maroon">c</font> <a href="aff_4.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">c</font> <a href="aff_4.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E24"><i><font color="Green">Lemma67</font></i></a>, <a class="txt" href="aff_4.html#E25"><i><font color="Green">Lemma68</font></i></a>, , , <a class="txt" href="aff_4.html#E4:81"><i><font color="Green">E37</font></i></a>, <a class="txt" href="aff_4.html#E1"><i><font color="Green">Lemma32</font></i></a></i>;<br/><a NAME="E7:81_1"/><b>then </b>
<font color="Maroon">c</font> <a href="aff_4.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E24"><i><font color="Green">Lemma67</font></i></a>, , , , , </i>;<br/><b>hence </b><a NAME="E8:81_1"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E24"><i><font color="Green">Lemma67</font></i></a>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:81"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T12">AFF_1:12</a></i>;<br/>


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