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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">p</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">q</font>, <font color="Olive" title="b5">o</font>, <font color="Olive" title="b6">c</font>, <font color="Olive" title="b7">r</font>, <font color="Olive" title="b8">d</font>, <font color="Olive" title="b9">s</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#R4" title="SEMI_AF1:pred.4">trap</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">q</font>,<font color="Olive" title="b5">o</font> &amp;  <a href="semi_af1.html#R4" title="SEMI_AF1:pred.4">trap</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">p</font>,<font color="Olive" title="b6">c</font>,<font color="Olive" title="b7">r</font>,<font color="Olive" title="b5">o</font> &amp;  <a href="semi_af1.html#R4" title="SEMI_AF1:pred.4">trap</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">q</font>,<font color="Olive" title="b8">d</font>,<font color="Olive" title="b9">s</font>,<font color="Olive" title="b5">o</font> holds <br/><font color="Olive" title="b6">c</font>,<font color="Olive" title="b8">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">r</font>,<font color="Olive" title="b9">s</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">q</font>, <font color="Maroon" title="c6">o</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">r</font>, <font color="Maroon" title="c9">d</font>, <font color="Maroon" title="c10">s</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#R4" title="SEMI_AF1:pred.4">trap</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">o</font> &amp;  <a href="semi_af1.html#R4" title="SEMI_AF1:pred.4">trap</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c6">o</font> &amp;  <a href="semi_af1.html#R4" title="SEMI_AF1:pred.4">trap</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c6">o</font> implies <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c9">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c10">s</font> )</span><br/>



















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:105"/><span class="lab"><font color="Green" title="E82">A1</font></span>: 
 <a href="semi_af1.html#R4" title="SEMI_AF1:pred.4">trap</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">o</font>
 <span class="kw">and </span><br/><a NAME="E2:105"/><span class="lab"><font color="Green" title="E83">A2</font></span>: 
 <a href="semi_af1.html#R4" title="SEMI_AF1:pred.4">trap</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c6">o</font>
 <span class="kw">and </span><br/><a NAME="E3:105"/><span class="lab"><font color="Green" title="E84">A3</font></span>: 
 <a href="semi_af1.html#R4" title="SEMI_AF1:pred.4">trap</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c6">o</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="c7">c</font>,<font color="Maroon" title="c9">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c10">s</font></span><br/>

<div><span class="lab"><font color="Green" title="E85">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:105_1"/><span class="lab"><font color="Green" title="E86">A5</font></span>: 
 not <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c9">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="c7">c</font>,<font color="Maroon" title="c9">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c10">s</font></span><br/><a NAME="E2:105_1"/>
 <a href="semi_af1.html#R4" title="SEMI_AF1:pred.4">trap</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="semi_af1.html#T93" target="_self" title="SEMI_AF1:th.93">Th121</a></span>;<br/><a NAME="E3:105_1"/><span class="kw">then </span>
 <a href="semi_af1.html#R4" title="SEMI_AF1:pred.4">trap</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c6">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:105"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="txt" href="#E1:105_1"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="ref" href="semi_af1.html#T99" target="_self" title="SEMI_AF1:th.99">Th127</a></span>;<br/><span class="kw">hence </span><a NAME="E4:105_1"/>
<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c9">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c10">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="ref" href="semi_af1.html#T98" target="_self" title="SEMI_AF1:th.98">Th126</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="E86">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:105_2"/>
 not <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</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="#E1:105"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="semi_af1.html#D8" target="_self" title="SEMI_AF1:def.8">Def8</a></span>;<br/><a NAME="E2:105_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">o</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="ref" href="semi_af1.html#T22" target="_self" title="SEMI_AF1:th.22">Th37</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">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="E4:105_2"/><span class="lab"><font color="Green" title="E87">A7</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="c2">a</font>,<font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T44" target="_self" title="SEMI_AF1:th.44">Th62</a></span>;<br/><a NAME="E5:105_2"/>
<font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">q</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="#E1:105"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="semi_af1.html#D8" target="_self" title="SEMI_AF1:def.8">Def8</a></span>;<br/><a NAME="E6:105_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</a></span>;<br/><a NAME="E7:105_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A8</font></span>: 
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">o</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T6" target="_self" title="SEMI_AF1:th.6">Th17</a></span>;<br/><a NAME="E8:105_2"/><span class="lab"><font color="Green" title="E89">A9</font></span>: 
<font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c11">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T1" target="_self" title="SEMI_AF1:th.1">Th12</a></span>;<br/><a NAME="E9:105_2"/>
( <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">o</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:105_2"><span class="lab"><font color="Green" title="E87">A7</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="E10:105_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A10</font></span>: 
<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:105_2"><span class="lab"><font color="Green" title="E88">A8</font></span></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></span>;<br/><a NAME="E11:105_2"/><span class="lab"><font color="Green" title="E91">A11</font></span>: 
 not <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c4">b</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="#E4:105_2"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="ref" href="semi_af1.html#T38" target="_self" title="SEMI_AF1:th.38">Th56</a></span>;<br/><a NAME="E12:105_2"/><span class="lab"><font color="Green" title="E92">A12</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:105"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="ref" href="semi_af1.html#T90" target="_self" title="SEMI_AF1:th.90">Th118</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E13:105_2"/><span class="lab"><font color="Green" title="E93">A13</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font>
 <span class="kw">and </span><br/><a NAME="E14:105_2"/><span class="lab"><font color="Green" title="E94">A14</font></span>: 
<font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">c</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E15:105_2"/><span class="lab"><font color="Green" title="E95">A15</font></span>: 
<font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c9">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="c7">c</font>,<font color="Maroon" title="c9">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c10">s</font></span><br/><a NAME="E16:105_2"/>
 not <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">q</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="#E1:105"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E13:105_2"><span class="lab"><font color="Green" title="E93">A13</font></span></a>, <a class="ref" href="semi_af1.html#T95" target="_self" title="SEMI_AF1:th.95">Th123</a></span>;<br/><a NAME="E17:105_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">o</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="ref" href="semi_af1.html#T22" target="_self" title="SEMI_AF1:th.22">Th37</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">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="E19:105_2"/><span class="lab"><font color="Green" title="E96">A16</font></span>: 
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c12">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T44" target="_self" title="SEMI_AF1:th.44">Th62</a></span>;<br/><a NAME="E20:105_2"/><span class="lab"><font color="Green" title="E97">A17</font></span>: 
 not <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c2">a</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="#E4:105_2"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="ref" href="semi_af1.html#T38" target="_self" title="SEMI_AF1:th.38">Th56</a></span>;<br/><a NAME="E21:105_2"/><span class="lab"><font color="Green" title="E98">A18</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="ref" href="semi_af1.html#T90" target="_self" title="SEMI_AF1:th.90">Th118</a></span>;<br/><a NAME="E22:105_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="c3">p</font>,<font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="semi_af1.html#D8" target="_self" title="SEMI_AF1:def.8">Def8</a></span>;<br/><a NAME="E23:105_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A19</font></span>: 
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T6" target="_self" title="SEMI_AF1:th.6">Th17</a></span>;<br/><a NAME="E24:105_2"/><span class="lab"><font color="Green" title="E100">A20</font></span>: 
<font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">p</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="#E1:105"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="semi_af1.html#D8" target="_self" title="SEMI_AF1:def.8">Def8</a></span>;<br/><a NAME="E25:105_2"/>
( <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:105_2"><span class="lab"><font color="Green" title="E87">A7</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="E26:105_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A21</font></span>: 
<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:105_2"><span class="lab"><font color="Green" title="E99">A19</font></span></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></span>;<br/><a NAME="E27:105_2"/><span class="lab"><font color="Green" title="E102">A22</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:105_2"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="ref" href="semi_af1.html#T36" target="_self" title="SEMI_AF1:th.36">Th54</a></span>;<br/><a NAME="E28:105_2"/>
( <font color="Maroon" title="c5">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c12">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:105_2"><span class="lab"><font color="Green" title="E96">A16</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="E29:105_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c11">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c12">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:105_2"><span class="lab"><font color="Green" title="E101">A21</font></span></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></span>;<br/><a NAME="E30:105_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A23</font></span>: 
<font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</a></span>;<br/><a NAME="E31:105_2"/>
( <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c12">y</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">o</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:105_2"><span class="lab"><font color="Green" title="E96">A16</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="E32:105_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A24</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c11">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c12">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:105_2"><span class="lab"><font color="Green" title="E90">A10</font></span></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></span>;<br/><a NAME="E33:105_2"/>
 not <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c11">x</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="#E4:105_2"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="ref" href="semi_af1.html#T38" target="_self" title="SEMI_AF1:th.38">Th56</a></span>;<br/><a NAME="E34:105_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A25</font></span>: 
 <a href="semi_af1.html#R4" title="SEMI_AF1:pred.4">trap</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c6">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:105_2"><span class="lab"><font color="Green" title="E103">A23</font></span></a>, <a class="txt" href="#E32:105_2"><span class="lab"><font color="Green" title="E104">A24</font></span></a>, <a class="txt" href="#E24:105_2"><span class="lab"><font color="Green" title="E100">A20</font></span></a>, <a class="ref" href="semi_af1.html#D8" target="_self" title="SEMI_AF1:def.8">Def8</a></span>;<br/><a NAME="E35:105_2"/>
 not <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c11">x</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="#E4:105_2"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="ref" href="semi_af1.html#T38" target="_self" title="SEMI_AF1:th.38">Th56</a></span>;<br/><a NAME="E36:105_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A26</font></span>: 
 <a href="semi_af1.html#R4" title="SEMI_AF1:pred.4">trap</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c6">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E34:105_2"><span class="lab"><font color="Green" title="E105">A25</font></span></a>, <a class="ref" href="semi_af1.html#T99" target="_self" title="SEMI_AF1:th.99">Th127</a></span>;<br/><a NAME="E37:105_2"/>
<font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c9">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:105_2"><span class="lab"><font color="Green" title="E95">A15</font></span></a>, <a class="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</a></span>;<br/><a NAME="E38:105_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A27</font></span>: 
 <a href="semi_af1.html#R4" title="SEMI_AF1:pred.4">trap</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c6">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:105"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="txt" href="#E36:105_2"><span class="lab"><font color="Green" title="E106">A26</font></span></a>, <a class="txt" href="#E27:105_2"><span class="lab"><font color="Green" title="E102">A22</font></span></a>, <a class="txt" href="#E12:105_2"><span class="lab"><font color="Green" title="E92">A12</font></span></a>, <a class="txt" href="#E20:105_2"><span class="lab"><font color="Green" title="E97">A17</font></span></a>, <a class="txt" href="#E8:105_2"><span class="lab"><font color="Green" title="E89">A9</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#T99" target="_self" title="SEMI_AF1:th.99">Th127</a></span>;<br/><a NAME="E39:105_2"/>
<font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c7">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:105_2"><span class="lab"><font color="Green" title="E94">A14</font></span></a>, <a class="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</a></span>;<br/><a NAME="E40:105_2"/><span class="kw">then </span>
 <a href="semi_af1.html#R4" title="SEMI_AF1:pred.4">trap</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c6">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E34:105_2"><span class="lab"><font color="Green" title="E105">A25</font></span></a>, <a class="txt" href="#E11:105_2"><span class="lab"><font color="Green" title="E91">A11</font></span></a>, <a class="txt" href="#E27:105_2"><span class="lab"><font color="Green" title="E102">A22</font></span></a>, <a class="txt" href="#E21:105_2"><span class="lab"><font color="Green" title="E98">A18</font></span></a>, <a class="txt" href="#E8:105_2"><span class="lab"><font color="Green" title="E89">A9</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#T99" target="_self" title="SEMI_AF1:th.99">Th127</a></span>;<br/><span class="kw">hence </span><a NAME="E41:105_2"/>
<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c9">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c10">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:105_2"><span class="lab"><font color="Green" title="E107">A27</font></span></a>, <a class="ref" href="semi_af1.html#T98" target="_self" title="SEMI_AF1:th.98">Th126</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="E87">A28</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:105_3"/><span class="lab"><font color="Green" title="E88">A29</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</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="c7">c</font>,<font color="Maroon" title="c9">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c10">s</font></span><br/><a NAME="E2:105_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="semi_af1.html#T93" target="_self" title="SEMI_AF1:th.93">Th121</a>, <a class="ref" href="semi_af1.html#T94" target="_self" title="SEMI_AF1:th.94">Th122</a></span>;<br/><a NAME="E3:105_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A30</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:105"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="ref" href="semi_af1.html#T93" target="_self" title="SEMI_AF1:th.93">Th121</a>, <a class="ref" href="semi_af1.html#T94" target="_self" title="SEMI_AF1:th.94">Th122</a></span>;<br/><a NAME="E4:105_3"/>
<font color="Maroon" title="c6">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E1:105_3"><span class="lab"><font color="Green" title="E88">A29</font></span></a>, <a class="ref" href="semi_af1.html#T93" target="_self" title="SEMI_AF1:th.93">Th121</a>, <a class="ref" href="semi_af1.html#T94" target="_self" title="SEMI_AF1:th.94">Th122</a></span>;<br/><span class="kw">hence </span><a NAME="E5:105_3"/>
<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c9">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c10">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:105_3"><span class="lab"><font color="Green" title="E89">A30</font></span></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</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="105_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:105_4"/>
 not <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">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="c7">c</font>,<font color="Maroon" title="c9">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c10">s</font></span><br/><a NAME="E2:105_4"/><span class="kw">then </span>
 <a href="semi_af1.html#R4" title="SEMI_AF1:pred.4">trap</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c6">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:105"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="ref" href="semi_af1.html#T99" target="_self" title="SEMI_AF1:th.99">Th127</a></span>;<br/><span class="kw">hence </span><a NAME="E3:105_4"/>
<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c9">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c10">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:105"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="ref" href="semi_af1.html#T98" target="_self" title="SEMI_AF1:th.98">Th126</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="E8:105"/>
<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c9">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c10">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:105"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E6:105"><span class="lab"><font color="Green" title="E87">A28</font></span></a>, <a class="txt" href="#E5:105"><span class="lab"><font color="Green" title="E86">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/>


</div>
