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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>assume </b><a NAME="E1:37"/><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">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 ;<br/>

<a NAME="E2:37"/><b>then </b>
not <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> 
 
<b>by </b><i><a class="txt" href="semi_af1.html#E1:37"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:37"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>

<b>assume </b><a NAME="E4:37"/><i><font color="Green">E30</font></i>: 
(  not <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> or  not <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> or  not <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> )
 ;<br/>

<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 ;<br/><a NAME="E2:37_1"/><b>then </b>
( not <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; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">a</font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E5:37"><i><font color="Green">E31</font></i></a>, <a class="txt" href="semi_af1.html#E1:37"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E3:37_1"/><b>then </b>
( not <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; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:37_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 ;<br/><a NAME="E2:37_2"/><b>then </b>
( not <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; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E5:37"><i><font color="Green">E31</font></i></a>, <a class="txt" href="semi_af1.html#E1:37"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E3:37_2"/><b>then </b>
( not <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; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> )
 <b>by </b><i/>;<br/><a NAME="E4:37_2"/><b>then </b>
( not <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; <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:37_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_3"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 ;<br/><a NAME="E2:37_3"/><b>then </b>
( not <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; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E5:37"><i><font color="Green">E31</font></i></a>, <a class="txt" href="semi_af1.html#E1:37"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E3:37_3"/><b>then </b>
( not <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; <font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">b</font> )
 <b>by </b><i/>;<br/><a NAME="E4:37_3"/><b>then </b>
( not <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; <font color="Maroon">c</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/>;<br/><b>hence </b><a NAME="E5:37_3"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:37"/>
contradiction
 <b>by </b><i><a class="txt" href="semi_af1.html#E6:37"><i><font color="Green">E32</font></i></a>, , </i>;<br/>


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