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<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">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font>, <font color="Maroon" title="c6">r</font>, <font color="Maroon" title="c7">s</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="E1:64"/><i><font color="Green" title="E54">A1</font></i>: 
(  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</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="c6">r</font>,<font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_1"/><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">q</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="E3:64_1"/><i><font color="Green" title="E56">A3</font></i>: 
(  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> &amp;  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:64"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</a></i>;<br/><a NAME="E4:64_1"/><i><font color="Green" title="E57">A4</font></i>: 
not <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c4">c</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#E1:64"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="semi_af1.html#T57" target="_self" title="SEMI_AF1:th.57">Th57</a></i>;<br/><a NAME="E5:64_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:64"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="semi_af1.html#E1:64_1"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="semi_af1.html#D3" target="_self" title="SEMI_AF1:def.3">Def3</a>, <a class="ref" href="semi_af1.html#T76" target="_self" title="SEMI_AF1:th.76">Th76</a></i>;<br/><a NAME="E6:64_1"/><b>then </b><i><font color="Green" title="E58">A5</font></i>: 
<font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T20" target="_self" title="SEMI_AF1:th.20">Th20</a></i>;<br/><a NAME="E7:64_1"/>
(  <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">b</font>,<font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</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">b</font>,<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font> &amp;  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:64"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="semi_af1.html#E3:64_1"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="semi_af1.html#T61" target="_self" title="SEMI_AF1:th.61">Th61</a></i>;<br/><a NAME="E8:64_1"/><b>then </b>
<font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T70" target="_self" title="SEMI_AF1:th.70">Th70</a></i>;<br/><b>hence </b><a NAME="E9:64_1"/>
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E4:64_1"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="semi_af1.html#E6:64_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="semi_af1.html#D3" target="_self" title="SEMI_AF1:def.3">Def3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:64"/>
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:64"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="semi_af1.html#T54" target="_self" title="SEMI_AF1:th.54">Th54</a></i>;<br/>


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