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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">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font>  ex <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#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/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</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">  ex <font color="Olive" title="b1">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="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font></span><br/>







<div><span class="lab"><font color="Green" title="E56">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:85_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font></span><br/><a NAME="E2:85_1"/><span class="kw">then </span>
 <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="c4">c</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>;<br/><span class="kw">hence </span><a NAME="E3:85_1"/>
 ex <font color="Olive" title="b1">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="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font>
 ; <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="E57">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:85_2"/><span class="lab"><font color="Green" title="E58">A3</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">b</font>
 <span class="kw">and </span><br/><a NAME="E2:85_2"/><span class="lab"><font color="Green" title="E59">A4</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">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="E4:85_2"/><span class="lab"><font color="Green" title="E60">A5</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">b</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85_2"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="semi_af1.html#E74"><span class="lab"><font color="Green" title="E44">Lm1</font></span></a></span>;<br/><a NAME="E5:85_2"/>
not <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</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">by</span> <span class="lab"><a class="txt" href="#E2:85_2"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E4:85_2"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="ref" href="semi_af1.html#T57" target="_self" title="SEMI_AF1:th.57">Th57</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">d</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="E7:85_2"/><span class="lab"><font color="Green" title="E61">A6</font></span>: 
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T62" target="_self" title="SEMI_AF1:th.62">Th62</a></span>;<br/><a NAME="E8:85_2"/>
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:85_2"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="ref" href="semi_af1.html#T61" target="_self" title="SEMI_AF1:th.61">Th61</a></span>;<br/><a NAME="E9:85_2"/><span class="kw">then </span>
 <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="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:85_2"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E10:85_2"/>
 ex <font color="Olive" title="b1">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="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font>
 ; <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="85_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:85_3"/>
<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>
 <span class="kw">and </span><br/><a NAME="E2:85_3"/><span class="lab"><font color="Green" title="E58">A7</font></span>: 
not <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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font></span><br/><a NAME="E3:85_3"/>
 ex <font color="Olive" title="b1">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="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:85_3"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="ref" href="semi_af1.html#T62" target="_self" title="SEMI_AF1:th.62">Th62</a></span>;<br/><span class="kw">hence </span><a NAME="E4:85_3"/>
 ex <font color="Olive" title="b1">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="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T79" target="_self" title="SEMI_AF1:th.79">Th79</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="E4:85"/>
 ex <font color="Olive" title="b1">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="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E2:85"><span class="lab"><font color="Green" title="E57">A2</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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