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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><b>that </b><br/><a NAME="E1:45"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E2:45"/><i><font color="Green">E35</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>  ) or ( <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/>

<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:45_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">X</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>and </b><br/><a NAME="E2:45_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">Y</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 ;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E4:45_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E5:45_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E9"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><a NAME="E6:45_1"/>
( <font color="Maroon">a</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">X</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E5:45_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/><b>hence </b><a NAME="E7:45_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T3" target="_self">Th3</a>, <a class="txt" href="aff_4.html#E2:45_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="aff_4.html#E4:45_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="aff_4.html#E5:45_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="aff_1.html#T71">AFF_1:71</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>now </b></a><div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:45_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>and </b><br/><a NAME="E2:45_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">Y</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 ;<br/><b>consider </b><font color="Maroon">K</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><b> such that </b><br/><a NAME="E4:45_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">P</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; not <font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font> )
 <b>and </b><br/><a NAME="E5:45_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E9"><i><font color="Green">Lemma42</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E7:45_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E8:45_2"/>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T31">AFF_1:31</a></i>;<br/><b>set </b><font color="Maroon">M</font> = <font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">K</font>;<br/><a NAME="E9:45_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">K</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" target="_self">*</a> <font color="Maroon">K</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E10:45_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E11:45_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i>, </i>;<br/><a NAME="E12:45_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E13:45_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E14:45_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="aff_4.html#T8" target="_self">Th8</a>, <a class="txt" href="aff_4.html#E1:45_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="aff_4.html#E2:45_2"><i><font color="Green">E49</font></i></a>, </i>;<br/><a NAME="E15:45_2"/><b>then </b>
<font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</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#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E16:45_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E9"><i><font color="Green">Lemma42</font></i></a>, , , <a class="txt" href="aff_4.html#E1:45_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="aff_4.html#T9" target="_self">Th9</a>, <a class="ref" href="aff_4.html#T10" target="_self">Th10</a>, <a class="ref" href="aff_4.html#T11" target="_self">Th11</a>, <a class="ref" href="aff_4.html#T12" target="_self">Th12</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:45"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T4" target="_self">Th4</a>, <a class="ref" href="aff_4.html#T7" target="_self">Th7</a></i>;<br/>


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