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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">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">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">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">d'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/>

















<b>assume </b><a NAME="E1:53"/><i><font color="Green">E29</font></i>: 
(  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font>,<font color="Maroon">b</font>,<font color="Maroon">b'</font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font>,<font color="Maroon">d</font>,<font color="Maroon">d'</font> )
 ;<br/>

<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_1"/>
not <font color="Maroon">b</font>,<font color="Maroon">b'</font>,<font color="Maroon">c</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 ;<br/><a NAME="E2:53_1"/><b>then </b>
(  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font>,<font color="Maroon">d</font>,<font color="Maroon">d'</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:53_1"/>
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c'</font>,<font color="Maroon">d'</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_2"/>
not <font color="Maroon">a</font>,<font color="Maroon">a'</font>,<font color="Maroon">d</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 ;<br/><a NAME="E2:53_2"/><b>then </b>
( not <font color="Maroon">a</font>,<font color="Maroon">a'</font>,<font color="Maroon">d</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">b</font>,<font color="Maroon">b'</font>,<font color="Maroon">a</font>,<font color="Maroon">a'</font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font>,<font color="Maroon">d</font>,<font color="Maroon">d'</font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:53_2"/><b>then </b>
(  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font>,<font color="Maroon">d</font>,<font color="Maroon">d'</font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:53_2"/>
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c'</font>,<font color="Maroon">d'</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_3"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">b</font>,<font color="Maroon">b'</font>,<font color="Maroon">c</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">a</font>,<font color="Maroon">a'</font>,<font color="Maroon">d</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  )
 ;<br/><a NAME="E2:53_3"/>
<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="txt" href="semi_af1.html#E2:53"><i><font color="Green">E30</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font><b> such that </b><br/><a NAME="E4:53_3"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/><a NAME="E5:53_3"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T14" target="_self">Th14</a>, </i>;<br/><a NAME="E6:53_3"/><i><font color="Green">E90</font></i>: 
not <font color="Maroon">a</font>,<font color="Maroon">a'</font>,<font color="Maroon">b</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i>, <a class="txt" href="semi_af1.html#E1:53_3"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E7:53_3"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">a'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font>
 <b>by </b><i/>;<br/><a NAME="E8:53_3"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a'</font>
 <b>by </b><i>, <a class="txt" href="semi_af1.html#E2:53"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E9:53_3"/><b>then </b>
not <font color="Maroon">a</font>,<font color="Maroon">a'</font>,<font color="Maroon">x</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i><a class="ref" href="semi_af1.html#T14" target="_self">Th14</a>, <a class="ref" href="semi_af1.html#T15" target="_self">Th15</a>, <a class="ref" href="semi_af1.html#T16" target="_self">Th16</a>, <a class="ref" href="semi_af1.html#T17" target="_self">Th17</a>, </i>;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font><b> such that </b><br/><a NAME="E11:53_3"/><i><font color="Green">E92</font></i>: 
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i/>;<br/><a NAME="E12:53_3"/><i><font color="Green">E93</font></i>: 
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="semi_af1.html#T14" target="_self">Th14</a>, <a class="ref" href="semi_af1.html#T18" target="_self">Th18</a>, </i>;<br/><a NAME="E13:53_3"/><b>then </b>
( not <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp; not <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">d</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="semi_af1.html#T13" target="_self">Th13</a>, <a class="ref" href="semi_af1.html#T18" target="_self">Th18</a>, </i>;<br/><a NAME="E14:53_3"/><b>then </b>
(  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">d</font>,<font color="Maroon">d'</font> )
 <b>by </b><i>, <a class="ref" href="semi_af1.html#T18" target="_self">Th18</a>, <a class="ref" href="semi_af1.html#T20" target="_self">Th20</a>, </i>;<br/><b>hence </b><a NAME="E15:53_3"/>
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c'</font>,<font color="Maroon">d'</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:53"/>
<font color="Maroon">c</font>,<font color="Maroon">d</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#T12" target="_self">Th12</a></i>;<br/>


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