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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font>  st  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> holds <br/> <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font></span><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> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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="c4">c</font>,<font color="Maroon" title="c5">d</font> implies  <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> )</span><br/>









<b>assume </b><a NAME="E1:63"/><i><font color="Green" title="E53">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">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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></span><br/>

<a NAME="E2:63"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font>
 
<b>by </b><i><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="c6">p</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="E4:63"/><i><font color="Green" title="E54">A2</font></i>: 
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">p</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">p</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="E5:63"/>
( not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</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="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">p</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="c2">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">p</font> &amp; <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> )
 
<b>by </b><i><a class="txt" href="semi_af1.html#E1:63"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="semi_af1.html#E4:63"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</a>, <a class="ref" href="semi_af1.html#T12" target="_self" title="SEMI_AF1:th.12">Th12</a>, <a class="ref" href="semi_af1.html#T54" target="_self" title="SEMI_AF1:th.54">Th54</a>, <a class="ref" href="semi_af1.html#T56" target="_self" title="SEMI_AF1:th.56">Th56</a></i>;<br/>
<a NAME="E6:63"/><b>then </b>
not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c3">b</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 
<b>by </b><i><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="c7">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="E8:63"/><i><font color="Green" title="E55">A3</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="c6">p</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">q</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="E9:63"/>
 <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">p</font>,<font color="Maroon" title="c7">q</font>
 
<b>by </b><i><a class="txt" href="semi_af1.html#E8:63"><i><font color="Green" title="E55">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="E10:63"/><b>then </b>
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font>
 
<b>by </b><i><a class="txt" href="semi_af1.html#E1:63"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="semi_af1.html#E4:63"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="semi_af1.html#T69" target="_self" title="SEMI_AF1:th.69">Th69</a></i>;<br/>
<a NAME="E11:63"/><b>then </b>
(  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">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="c6">p</font>,<font color="Maroon" title="c7">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#E8:63"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="semi_af1.html#T61" target="_self" title="SEMI_AF1:th.61">Th61</a></i>;<br/>
<b>hence </b><a NAME="E12:63"/>
 <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>
 <b>by </b><i><a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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