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

<b>let </b><font color="Maroon" title="c1">SAS</font> be   <a href="semi_af1.html#NM1" title="SEMI_AF1:NM.1">Semi_Affine_Space</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">a'</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">b'</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">c'</font>, <font color="Maroon" title="c8">d</font>, <font color="Maroon" title="c9">d'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font>;<br/>

















<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E44">A1</font></i>: 
(  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b'</font> &amp;  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c'</font> &amp;  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c9">d'</font> )
 ;<br/>

<div><i><font color="Green" title="E45">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_1"/>
not <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c6">c</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 ;<br/><a NAME="E2:53_1"/><b>then </b>
(  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c'</font> &amp;  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c9">d'</font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:53"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="semi_af1.html#T68" target="_self" title="SEMI_AF1:th.68">Th68</a></i>;<br/><b>hence </b><a NAME="E3:53_1"/>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c9">d'</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T67" target="_self" title="SEMI_AF1:th.67">Th67</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E46">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_2"/>
not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c8">d</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 ;<br/><a NAME="E2:53_2"/><b>then </b>
( not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c8">d</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp;  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> &amp;  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c9">d'</font> &amp;  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c'</font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:53"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="semi_af1.html#T61" target="_self" title="SEMI_AF1:th.61">Th61</a></i>;<br/><a NAME="E3:53_2"/><b>then </b>
(  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c9">d'</font> &amp;  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c'</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T68" target="_self" title="SEMI_AF1:th.68">Th68</a></i>;<br/><b>hence </b><a NAME="E4:53_2"/>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c9">d'</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T67" target="_self" title="SEMI_AF1:th.67">Th67</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_3"/><i><font color="Green" title="E47">A4</font></i>: 
( <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c6">c</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c8">d</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  )
 ;<br/><a NAME="E2:53_3"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:53"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="semi_af1.html#T54" target="_self" title="SEMI_AF1:th.54">Th54</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font><b> such that </b><br/><a NAME="E4:53_3"/><i><font color="Green" title="E48">A5</font></i>: 
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c10">x</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T66" target="_self" title="SEMI_AF1:th.66">Th66</a></i>;<br/><a NAME="E5:53_3"/><i><font color="Green" title="E49">A6</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E4:53_3"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</a></i>;<br/><a NAME="E6:53_3"/><i><font color="Green" title="E50">A7</font></i>: 
not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c4">b</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:53"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="semi_af1.html#T56" target="_self" title="SEMI_AF1:th.56">Th56</a></i>;<br/><a NAME="E7:53_3"/><i><font color="Green" title="E51">A8</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T12" target="_self" title="SEMI_AF1:th.12">Th12</a></i>;<br/><a NAME="E8:53_3"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a'</font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:53"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="semi_af1.html#T54" target="_self" title="SEMI_AF1:th.54">Th54</a></i>;<br/><a NAME="E9:53_3"/><b>then </b>
not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c10">x</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="semi_af1.html#E4:53_3"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="semi_af1.html#E5:53_3"><i><font color="Green" title="E49">A6</font></i></a>, <a class="txt" href="semi_af1.html#E6:53_3"><i><font color="Green" title="E50">A7</font></i></a>, <a class="txt" href="semi_af1.html#E7:53_3"><i><font color="Green" title="E51">A8</font></i></a>, <a class="ref" href="semi_af1.html#T39" target="_self" title="SEMI_AF1:th.39">Th39</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font><b> such that </b><br/><a NAME="E11:53_3"/><i><font color="Green" title="E52">A9</font></i>: 
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T62" target="_self" title="SEMI_AF1:th.62">Th62</a></i>;<br/><a NAME="E12:53_3"/><i><font color="Green" title="E53">A10</font></i>: 
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:53"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="semi_af1.html#E4:53_3"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="semi_af1.html#E11:53_3"><i><font color="Green" title="E52">A9</font></i></a>, <a class="ref" href="semi_af1.html#T69" target="_self" title="SEMI_AF1:th.69">Th69</a></i>;<br/><a NAME="E13:53_3"/>
( not <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c6">c</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c8">d</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:53"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="semi_af1.html#E1:53_3"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="semi_af1.html#E11:53_3"><i><font color="Green" title="E52">A9</font></i></a>, <a class="ref" href="semi_af1.html#T57" target="_self" title="SEMI_AF1:th.57">Th57</a>, <a class="ref" href="semi_af1.html#T69" target="_self" title="SEMI_AF1:th.69">Th69</a></i>;<br/><a NAME="E14:53_3"/><b>then </b>
(  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c'</font> &amp;  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c9">d'</font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:53"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="semi_af1.html#E11:53_3"><i><font color="Green" title="E52">A9</font></i></a>, <a class="txt" href="semi_af1.html#E12:53_3"><i><font color="Green" title="E53">A10</font></i></a>, <a class="ref" href="semi_af1.html#T68" target="_self" title="SEMI_AF1:th.68">Th68</a></i>;<br/><b>hence </b><a NAME="E15:53_3"/>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c9">d'</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T67" target="_self" title="SEMI_AF1:th.67">Th67</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:53"/>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c9">d'</font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E2:53"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="semi_af1.html#E3:53"><i><font color="Green" title="E46">A3</font></i></a></i>;<br/>


</div>
