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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">b9</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">a9</font>, <font color="Olive" title="b6">c9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font>  st  not <font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">b9</font>,<font color="Olive" title="b3">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="Olive" title="b4">a</font>,<font color="Olive" title="b5">a9</font>,<font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">b9</font> &amp;  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">a9</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b6">c9</font> holds <br/> <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">b9</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b6">c9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">b9</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">a9</font>, <font color="Maroon" title="c7">c9</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">b9</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="c5">a</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">b9</font> &amp;  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">c9</font> implies  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">b9</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">c9</font> )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:51"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
 not <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">b9</font>,<font color="Maroon" title="c4">c</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E2:51"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">b9</font>
 <span class="kw">and </span><br/><a NAME="E3:51"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">b9</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">c9</font></span><br/>

<a NAME="E4:51"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="semi_af1.html#D3" target="_self" title="SEMI_AF1:def.3">Def3</a></span>;<br/>
<a NAME="E5:51"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a9</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">b9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E43">A2</font></span></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#T36" target="_self" title="SEMI_AF1:th.36">Th54</a></span>;<br/>
<a NAME="E6:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A5</font></span>: 
<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></span>;<br/>
<a NAME="E7:51"/>
<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b9</font>,<font color="Maroon" title="c7">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="semi_af1.html#T49" target="_self" title="SEMI_AF1:th.49">Th67</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:51"/>
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">b9</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E6:51"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="ref" href="semi_af1.html#D3" target="_self" title="SEMI_AF1:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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