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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="E1:63"/><i><font color="Green">E53</font></i>: 
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <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>
 ;<br/>

<a NAME="E2:63"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="semi_af1.html#T54" target="_self">Th54</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:63"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T66" target="_self">Th66</a></i>;<br/>
<a NAME="E5:63"/>
( not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</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>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</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>2</sub></font>,<font color="Maroon">c<sub>3</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>6</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> )
 
<b>by </b><i><a class="txt" href="semi_af1.html#E1:63"><i><font color="Green">E53</font></i></a>, <a class="txt" href="semi_af1.html#E4:63"><i><font color="Green">E54</font></i></a>, <a class="ref" href="semi_af1.html#D2" target="_self">Def2</a>, <a class="ref" href="semi_af1.html#T12" target="_self">Th12</a>, <a class="ref" href="semi_af1.html#T54" target="_self">Th54</a>, <a class="ref" href="semi_af1.html#T56" target="_self">Th56</a></i>;<br/>
<a NAME="E6:63"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="semi_af1.html#T39" target="_self">Th39</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:63"/><i><font color="Green">E55</font></i>: 
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T62" target="_self">Th62</a></i>;<br/>
<a NAME="E9:63"/>
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="semi_af1.html#E8:63"><i><font color="Green">E55</font></i></a>, <a class="ref" href="semi_af1.html#T61" target="_self">Th61</a></i>;<br/>
<a NAME="E10:63"/><b>then </b>
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="semi_af1.html#E1:63"><i><font color="Green">E53</font></i></a>, <a class="txt" href="semi_af1.html#E4:63"><i><font color="Green">E54</font></i></a>, <a class="ref" href="semi_af1.html#T69" target="_self">Th69</a></i>;<br/>
<a NAME="E11:63"/><b>then </b>
(  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="semi_af1.html#E8:63"><i><font color="Green">E55</font></i></a>, <a class="ref" href="semi_af1.html#T61" target="_self">Th61</a></i>;<br/>
<b>hence </b><a NAME="E12:63"/>
 <a href="semi_af1.html#R3" target="_self">congr</a> <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>
 <b>by </b><i><a class="ref" href="semi_af1.html#D4" target="_self">Def4</a></i>;<br/>


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