<?xml version="1.0"?>
<div class="add">

<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> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:66_1"/><b>then </b>
 <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>4</sub></font>
 <b>by </b><i><a class="ref" href="semi_af1.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E3:66_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st  <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="Olive">b<sub>1</sub></font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2"/><i><font color="Green">E58</font></i>: 
( <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>,<font color="Maroon">c<sub>4</sub></font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <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="E3:66_2"/><i><font color="Green">E59</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>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E74"><i><font color="Green">Lemma44</font></i></a></i>;<br/><a NAME="E4:66_2"/><i><font color="Green">E60</font></i>: 
(  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:66_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="semi_af1.html#E3:66_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="semi_af1.html#T57" target="_self">Th57</a>, <a class="ref" href="semi_af1.html#T61" target="_self">Th61</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="E6:66_2"/><i><font color="Green">E61</font></i>: 
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</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="E7:66_2"/>
 <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>7</sub></font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E4:66_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="semi_af1.html#E6:66_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="semi_af1.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E8:66_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st  <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="Olive">b<sub>1</sub></font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_3"/>
( <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; 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>  )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</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="E3:66_3"/><i><font color="Green">E58</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>
 <b>by </b><i><a class="ref" href="semi_af1.html#T62" target="_self">Th62</a></i>;<br/><a NAME="E4:66_3"/>
 <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="txt" href="semi_af1.html#E3:66_3"><i><font color="Green">E58</font></i></a>, <a class="ref" href="semi_af1.html#T79" target="_self">Th79</a></i>;<br/><b>hence </b><a NAME="E5:66_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st  <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="Olive">b<sub>1</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:66"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st  <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="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:66"><i><font color="Green">E56</font></i></a>, <a class="txt" href="semi_af1.html#E2:66"><i><font color="Green">E57</font></i></a></i>;<br/>


</div>
