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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">a</font>, <font color="Olive" title="b2">a9</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">b9</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">c9</font>, <font color="Olive" title="b7">d</font>, <font color="Olive" title="b8">d9</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">a9</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">b9</font> &amp;  <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">a9</font>,<font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">c9</font> &amp;  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">b9</font>,<font color="Olive" title="b7">d</font>,<font color="Olive" title="b8">d9</font> holds <br/><font color="Olive" title="b5">c</font>,<font color="Olive" title="b7">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">c9</font>,<font color="Olive" title="b8">d9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">a9</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">b9</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">c9</font>, <font color="Maroon" title="c8">d</font>, <font color="Maroon" title="c9">d9</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"> (  <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">a9</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b9</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">a9</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c9</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">b9</font>,<font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c9">d9</font> implies <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">c9</font>,<font color="Maroon" title="c9">d9</font> )</span><br/>

















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:53"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
 <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">a9</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b9</font>
 <span class="kw">and </span><br/><a NAME="E2:53"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
 <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">a9</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c9</font>
 <span class="kw">and </span><br/><a NAME="E3:53"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
 <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">b9</font>,<font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c9">d9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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">c9</font>,<font color="Maroon" title="c9">d9</font></span><br/>

<div><span class="lab"><font color="Green" title="E47">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:53_1"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
 not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c8">d</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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">c9</font>,<font color="Maroon" title="c9">d9</font></span><br/><a NAME="E2:53_1"/>
 <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">b9</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="semi_af1.html#T43" target="_self" title="SEMI_AF1:th.43">Th61</a></span>;<br/><a NAME="E3:53_1"/><span class="kw">then </span>
 <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">a9</font>,<font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c9">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="txt" href="#E1:53_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="semi_af1.html#T50" target="_self" title="SEMI_AF1:th.50">Th68</a></span>;<br/><span class="kw">hence </span><a NAME="E4:53_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">c9</font>,<font color="Maroon" title="c9">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="semi_af1.html#T49" target="_self" title="SEMI_AF1:th.49">Th67</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/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E48">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:53_2"/><span class="lab"><font color="Green" title="E49">A7</font></span>: 
(  not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c4">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="c3">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="semi_af1.html#T1" target="_self" title="SEMI_AF1:th.1">Th12</a>, <a class="ref" href="semi_af1.html#T38" target="_self" title="SEMI_AF1:th.38">Th56</a></span>;<br/><a NAME="E2:53_2"/><span class="lab"><font color="Green" title="E50">A8</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="semi_af1.html#T36" target="_self" title="SEMI_AF1:th.36">Th54</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:53_2"/><span class="lab"><font color="Green" title="E51">A9</font></span>: 
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b9</font>,<font color="Maroon" title="c6">c</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E4:53_2"/><span class="lab"><font color="Green" title="E52">A10</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c8">d</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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">c9</font>,<font color="Maroon" title="c9">d9</font></span><br/><a NAME="E5:53_2"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="semi_af1.html#T36" target="_self" title="SEMI_AF1:th.36">Th54</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E7:53_2"/><span class="lab"><font color="Green" title="E53">A11</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E8:53_2"/><span class="lab"><font color="Green" title="E54">A12</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E9:53_2"/><span class="lab"><font color="Green" title="E55">A13</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T48" target="_self" title="SEMI_AF1:th.48">Th66</a></span>;<br/><a NAME="E10:53_2"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:53_2"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E12:53_2"/><span class="lab"><font color="Green" title="E56">A14</font></span>: 
 <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">a9</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:53_2"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="txt" href="#E1:53_2"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E2:53_2"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="ref" href="semi_af1.html#T23" target="_self" title="SEMI_AF1:th.23">Th39</a>, <a class="ref" href="semi_af1.html#T44" target="_self" title="SEMI_AF1:th.44">Th62</a></span>;<br/><a NAME="E13:53_2"/><span class="lab"><font color="Green" title="E57">A15</font></span>: 
 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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53_2"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="txt" href="#E12:53_2"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="ref" href="semi_af1.html#T39" target="_self" title="SEMI_AF1:th.39">Th57</a></span>;<br/><a NAME="E14:53_2"/>
 <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">b9</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E7:53_2"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="txt" href="#E9:53_2"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E12:53_2"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="ref" href="semi_af1.html#T51" target="_self" title="SEMI_AF1:th.51">Th69</a></span>;<br/><a NAME="E15:53_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A16</font></span>: 
 <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">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="txt" href="#E13:53_2"><span class="lab"><font color="Green" title="E57">A15</font></span></a>, <a class="ref" href="semi_af1.html#T50" target="_self" title="SEMI_AF1:th.50">Th68</a></span>;<br/><a NAME="E16:53_2"/>
 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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E3:53_2"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="txt" href="#E7:53_2"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="txt" href="#E9:53_2"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E12:53_2"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="ref" href="semi_af1.html#T39" target="_self" title="SEMI_AF1:th.39">Th57</a>, <a class="ref" href="semi_af1.html#T51" target="_self" title="SEMI_AF1:th.51">Th69</a></span>;<br/><a NAME="E17:53_2"/><span class="kw">then </span>
 <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">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="txt" href="#E12:53_2"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="ref" href="semi_af1.html#T50" target="_self" title="SEMI_AF1:th.50">Th68</a></span>;<br/><span class="kw">hence </span><a NAME="E18:53_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">c9</font>,<font color="Maroon" title="c9">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:53_2"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="ref" href="semi_af1.html#T49" target="_self" title="SEMI_AF1:th.49">Th67</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/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:53_3"/>
 not <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b9</font>,<font color="Maroon" title="c6">c</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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">c9</font>,<font color="Maroon" title="c9">d9</font></span><br/><a NAME="E2:53_3"/><span class="kw">then </span>
 <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">b9</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="semi_af1.html#T50" target="_self" title="SEMI_AF1:th.50">Th68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:53_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">c9</font>,<font color="Maroon" title="c9">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="semi_af1.html#T49" target="_self" title="SEMI_AF1:th.49">Th67</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:53"/>
<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">c9</font>,<font color="Maroon" title="c9">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E48">A6</font></span></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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