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<b>let </b><font color="Maroon">SAS</font> be   <a href="semi_af1.html#NM1">Semi_Affine_Space</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/>













<b>assume </b><a NAME="E1:64"/><i><font color="Green">E29</font></i>: 
(  <a href="semi_af1.html#R3" target="_self">congr</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>, <font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font><b> such that </b><br/><a NAME="E3:64_1"/><i><font color="Green">E31</font></i>: 
(  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T25" target="_self">Th25</a>, </i>;<br/><a NAME="E4:64_1"/><i><font color="Green">E32</font></i>: 
not <font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">c</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i><a class="ref" href="semi_af1.html#T25" target="_self">Th25</a>, </i>;<br/><a NAME="E5:64_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">r</font>,<font color="Maroon">s</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <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="semi_af1.html#T25" target="_self">Th25</a>, <a class="ref" href="semi_af1.html#T26" target="_self">Th26</a>, <a class="txt" href="semi_af1.html#E1:64"><i><font color="Green">E29</font></i></a>, <a class="ref" href="semi_af1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E6:64_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">r</font>,<font color="Maroon">s</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="semi_af1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E7:64_1"/>
(  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font>,<font color="Maroon">s</font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T25" target="_self">Th25</a>, <a class="ref" href="semi_af1.html#T28" target="_self">Th28</a>, </i>;<br/><a NAME="E8:64_1"/><b>then </b>
<font color="Maroon">r</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">s</font>,<font color="Maroon">d</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:64_1"/>
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T29" target="_self">Th29</a>, <a class="ref" href="semi_af1.html#T30" target="_self">Th30</a>, <a class="txt" href="semi_af1.html#E1:64"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:64"/>
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T25" target="_self">Th25</a>, </i>;<br/>


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