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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="semi_af1.html#NM1">Semi_Affine_Space</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>









<b>assume </b><a NAME="E1:27"/><i><font color="Green">E25</font></i>: 
( not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  )
 ;<br/><a NAME="E2:27_1"/><i><font color="Green">E27</font></i>: 
( not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:27"><i><font color="Green">E25</font></i></a>, <a class="ref" href="semi_af1.html#D1" target="_self">Def1</a>, <a class="ref" href="semi_af1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E3:27_1"/><b>then </b><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#D1" target="_self">Def1</a>, <a class="ref" href="semi_af1.html#T14" target="_self">Th14</a>, <a class="ref" href="semi_af1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E4:27_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:27_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="semi_af1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E5:27_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E6:27_1"/><b>then </b>
( not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:27_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="semi_af1.html#E2:27_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="semi_af1.html#E3:27_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="semi_af1.html#D1" target="_self">Def1</a>, <a class="ref" href="semi_af1.html#T29" target="_self">Th29</a></i>;<br/><b>hence </b><a NAME="E7:27_1"/>
contradiction
 <b>by </b><i><a class="ref" href="semi_af1.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:27"/>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:27"><i><font color="Green">E25</font></i></a></i>;<br/>


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