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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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">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#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> &amp;  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font>,<font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">d</font> holds <br/> <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">d</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">r</font>, <font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp;  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> implies  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:88"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>
 <span class="kw">and </span><br/><a NAME="E2:88"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font></span><br/>

<div><span class="lab"><font color="Green" title="E60">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:88_1"/>
<font color="Maroon" title="c2">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">s</font>
 <span class="kw">and </span><br/><a NAME="E2:88_1"/><span class="lab"><font color="Green" title="E61">A4</font></span>: 
( not <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">a</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</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="Red">::  thesis: </font></span></a><span class="hide">  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font></span><br/><a NAME="E3:88_1"/>
(  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">a</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">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E2:88_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="semi_af1.html#T78" target="_self" title="SEMI_AF1:th.78">Th78</a></span>;<br/><span class="kw">hence </span><a NAME="E4:88_1"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E61">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:88_2"/><span class="lab"><font color="Green" title="E62">A6</font></span>: 
<font color="Maroon" title="c2">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">s</font>
 <span class="kw">and </span><br/><a NAME="E2:88_2"/><span class="lab"><font color="Green" title="E63">A7</font></span>: 
<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</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="Red">::  thesis: </font></span></a><span class="hide">  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font></span><br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:88_2"/><span class="lab"><font color="Green" title="E64">A8</font></span>: 
 <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">r</font>,<font color="Maroon" title="c3">s</font>
 <span class="kw">and </span><br/><a NAME="E5:88_2"/><span class="lab"><font color="Green" title="E65">A9</font></span>: 
 <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">a</font>,<font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E1:88_2"><span class="lab"><font color="Green" title="E62">A6</font></span></a>, <a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</a></span>;<br/><a NAME="E6:88_2"/>
 <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="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E2:88_2"><span class="lab"><font color="Green" title="E63">A7</font></span></a>, <a class="txt" href="#E4:88_2"><span class="lab"><font color="Green" title="E64">A8</font></span></a>, <a class="ref" href="semi_af1.html#T80" target="_self" title="SEMI_AF1:th.80">Th80</a></span>;<br/><span class="kw">hence </span><a NAME="E7:88_2"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:88_2"><span class="lab"><font color="Green" title="E65">A9</font></span></a>, <a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E62">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="88_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:88_3"/><span class="lab"><font color="Green" title="E63">A11</font></span>: 
<font color="Maroon" title="c2">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">s</font>
 <span class="kw">and </span><br/><a NAME="E2:88_3"/><span class="lab"><font color="Green" title="E64">A12</font></span>: 
<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">a</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="Red">::  thesis: </font></span></a><span class="hide">  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font></span><br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:88_3"/><span class="lab"><font color="Green" title="E65">A13</font></span>: 
 <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">r</font>,<font color="Maroon" title="c3">s</font>
 <span class="kw">and </span><br/><a NAME="E5:88_3"/><span class="lab"><font color="Green" title="E66">A14</font></span>: 
 <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="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E1:88_3"><span class="lab"><font color="Green" title="E63">A11</font></span></a>, <a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</a></span>;<br/><a NAME="E6:88_3"/>
 <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">a</font>,<font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:88_3"><span class="lab"><font color="Green" title="E64">A12</font></span></a>, <a class="txt" href="#E4:88_3"><span class="lab"><font color="Green" title="E65">A13</font></span></a>, <a class="ref" href="semi_af1.html#T80" target="_self" title="SEMI_AF1:th.80">Th80</a></span>;<br/><span class="kw">hence </span><a NAME="E7:88_3"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:88_3"><span class="lab"><font color="Green" title="E66">A14</font></span></a>, <a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="88_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:88_4"/>
<font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font></span><br/><a NAME="E2:88_4"/><span class="kw">then </span>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="semi_af1.html#T73" target="_self" title="SEMI_AF1:th.73">Th73</a></span>;<br/><span class="kw">hence </span><a NAME="E3:88_4"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:88"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:88"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="txt" href="#E4:88"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="txt" href="#E3:88"><span class="lab"><font color="Green" title="E60">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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