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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">p</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">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/><b>let </b><font color="Maroon">o</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">r</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">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/>



















<b>assume </b><a NAME="E1:106"/><i><font color="Green">E29</font></i>: 
(  <a href="semi_af1.html#R4" target="_self">trap</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">b</font>,<font color="Maroon">q</font>,<font color="Maroon">o</font> &amp;  <a href="semi_af1.html#R4" target="_self">trap</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">c</font>,<font color="Maroon">r</font>,<font color="Maroon">o</font> &amp;  <a href="semi_af1.html#R4" target="_self">trap</a> <font color="Maroon">b</font>,<font color="Maroon">q</font>,<font color="Maroon">d</font>,<font color="Maroon">s</font>,<font color="Maroon">o</font> )
 ;<br/>

<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:106_1"/>
not <font color="Maroon">o</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="E2:106_1"/><b>then </b>
(  <a href="semi_af1.html#R4" target="_self">trap</a> <font color="Maroon">b</font>,<font color="Maroon">q</font>,<font color="Maroon">d</font>,<font color="Maroon">s</font>,<font color="Maroon">o</font> &amp;  <a href="semi_af1.html#R4" target="_self">trap</a> <font color="Maroon">b</font>,<font color="Maroon">q</font>,<font color="Maroon">c</font>,<font color="Maroon">r</font>,<font color="Maroon">o</font> )
 <b>by </b><i>, <a class="ref" href="semi_af1.html#T66" target="_self">Th66</a></i>;<br/><b>hence </b><a NAME="E3:106_1"/>
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T65" target="_self">Th65</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="106_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:106_2"/>
not <font color="Maroon">o</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="E2:106_2"/><b>then </b>
( not <font color="Maroon">o</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#R4" target="_self">trap</a> <font color="Maroon">b</font>,<font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">o</font> )
 <b>by </b><i>, <a class="ref" href="semi_af1.html#T61" target="_self">Th61</a></i>;<br/><a NAME="E3:106_2"/><b>then </b>
(  <a href="semi_af1.html#R4" target="_self">trap</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">d</font>,<font color="Maroon">s</font>,<font color="Maroon">o</font> &amp;  <a href="semi_af1.html#R4" target="_self">trap</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">c</font>,<font color="Maroon">r</font>,<font color="Maroon">o</font> )
 <b>by </b><i>, <a class="ref" href="semi_af1.html#T66" target="_self">Th66</a></i>;<br/><b>hence </b><a NAME="E4:106_2"/>
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T65" target="_self">Th65</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:106_3"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:106_3"/><b>then </b>
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 <b>by </b><i>, <a class="ref" href="semi_af1.html#T64" target="_self">Th64</a></i>;<br/><a NAME="E3:106_3"/><b>then </b>
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 <b>by </b><i>, <a class="ref" href="semi_af1.html#T64" target="_self">Th64</a></i>;<br/><b>hence </b><a NAME="E4:106_3"/>
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:106_4"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">o</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">o</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="E2:106_4"/><b>then </b>
( not <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp; not <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:106_4"><i><font color="Green">E84</font></i></a>, <a class="ref" href="semi_af1.html#T55" target="_self">Th55</a>, <a class="ref" href="semi_af1.html#T63" target="_self">Th63</a></i>;<br/><a NAME="E3:106_4"/><b>then </b><i><font color="Green">E89</font></i>: 
( not <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">o</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp; not <font color="Maroon">q</font>,<font color="Maroon">p</font>,<font color="Maroon">o</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  )
 <b>by </b><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="E5:106_4"/><i><font color="Green">E90</font></i>: 
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">o</font>,<font color="Maroon">x</font>
 <b>by </b><i/>;<br/><b>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="E7:106_4"/><i><font color="Green">E91</font></i>: 
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>,<font color="Maroon">o</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E3:106_4"><i><font color="Green">E89</font></i></a>, </i>;<br/><a NAME="E8:106_4"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:106_4"><i><font color="Green">E84</font></i></a>, <a class="ref" href="semi_af1.html#T55" target="_self">Th55</a></i>;<br/><a NAME="E9:106_4"/><b>then </b>
( <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">q</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E5:106_4"><i><font color="Green">E90</font></i></a>, <a class="txt" href="semi_af1.html#E7:106_4"><i><font color="Green">E91</font></i></a>, <a class="txt" href="semi_af1.html#E1:106"><i><font color="Green">E29</font></i></a>, , <a class="txt" href="semi_af1.html#E2:106"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E10:106_4"/><b>then </b>
( <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">q</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">q</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">y</font> )
 <b>by </b><i/>;<br/><a NAME="E11:106_4"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">y</font>
 <b>by </b><i/>;<br/><a NAME="E12:106_4"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i/>;<br/><a NAME="E13:106_4"/>
<font color="Maroon">o</font>,<font color="Maroon">b</font>,<font color="Maroon">q</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:106_4"><i><font color="Green">E84</font></i></a>, <a class="ref" href="semi_af1.html#T55" target="_self">Th55</a></i>;<br/><a NAME="E14:106_4"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">q</font>
 <b>by </b><i/>;<br/><a NAME="E15:106_4"/><b>then </b>
( <font color="Maroon">b</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">o</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">q</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E5:106_4"><i><font color="Green">E90</font></i></a>, <a class="txt" href="semi_af1.html#E7:106_4"><i><font color="Green">E91</font></i></a>, <a class="txt" href="semi_af1.html#E1:106"><i><font color="Green">E29</font></i></a>, , <a class="txt" href="semi_af1.html#E2:106"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E16:106_4"/><b>then </b>
( <font color="Maroon">q</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">q</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font> )
 <b>by </b><i/>;<br/><a NAME="E17:106_4"/><b>then </b>
( <font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp; not <font color="Maroon">o</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="txt" href="semi_af1.html#E1:106_4"><i><font color="Green">E84</font></i></a>, <a class="txt" href="semi_af1.html#E5:106_4"><i><font color="Green">E90</font></i></a>, , <a class="ref" href="semi_af1.html#T55" target="_self">Th55</a>, <a class="txt" href="semi_af1.html#E4:106"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E18:106_4"/><b>then </b><i><font color="Green">E93</font></i>: 
 <a href="semi_af1.html#R4" target="_self">trap</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E12:106_4"><i><font color="Green">E92</font></i></a>, <a class="ref" href="semi_af1.html#T55" target="_self">Th55</a></i>;<br/><a NAME="E19:106_4"/>
not <font color="Maroon">o</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i><a class="txt" href="semi_af1.html#E5:106_4"><i><font color="Green">E90</font></i></a>, <a class="txt" href="semi_af1.html#E4:106"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E20:106_4"/><b>then </b><i><font color="Green">E111</font></i>: 
 <a href="semi_af1.html#R4" target="_self">trap</a> <font color="Maroon">b</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:106_4"><i><font color="Green">E84</font></i></a>, <a class="txt" href="semi_af1.html#E18:106_4"><i><font color="Green">E93</font></i></a>, <a class="ref" href="semi_af1.html#T66" target="_self">Th66</a></i>;<br/><a NAME="E21:106_4"/>
( <font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">a</font> &amp; not <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">b</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; not <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">a</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:106_4"><i><font color="Green">E84</font></i></a>, <a class="ref" href="semi_af1.html#T70" target="_self">Th70</a>, <a class="txt" href="semi_af1.html#E5:106_4"><i><font color="Green">E90</font></i></a>, , , <a class="txt" href="semi_af1.html#E2:106"><i><font color="Green">E30</font></i></a>, <a class="txt" href="semi_af1.html#E4:106"><i><font color="Green">E32</font></i></a>, <a class="ref" href="semi_af1.html#T57" target="_self">Th57</a></i>;<br/><a NAME="E22:106_4"/><b>then </b>
( not <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">c</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp; not <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">d</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  )
 <b>by </b><i/>;<br/><a NAME="E23:106_4"/><b>then </b>
(  <a href="semi_af1.html#R4" target="_self">trap</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<font color="Maroon">r</font>,<font color="Maroon">o</font> &amp;  <a href="semi_af1.html#R4" target="_self">trap</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">d</font>,<font color="Maroon">s</font>,<font color="Maroon">o</font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:106_4"><i><font color="Green">E84</font></i></a>, <a class="txt" href="semi_af1.html#E18:106_4"><i><font color="Green">E93</font></i></a>, <a class="txt" href="semi_af1.html#E74"><i><font color="Green">Lemma94</font></i></a>, <a class="ref" href="semi_af1.html#T66" target="_self">Th66</a></i>;<br/><b>hence </b><a NAME="E24:106_4"/>
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T65" target="_self">Th65</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:106"/>
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T67" target="_self">Th67</a>, <a class="ref" href="semi_af1.html#T68" target="_self">Th68</a>, <a class="ref" href="semi_af1.html#T69" target="_self">Th69</a></i>;<br/>


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