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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">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="Red">::  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="E1:133"/><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="E2:133"/><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="E3:133"/><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="Red">::  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="133_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:133_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="Red">::  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="E2:133_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="#E1:133"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="semi_af1.html#T121" target="_self" title="SEMI_AF1:th.121">Th121</a></span>;<br/><a NAME="E3:133_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="#E3:133"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="txt" href="#E1:133_1"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="ref" href="semi_af1.html#T127" target="_self" title="SEMI_AF1:th.127">Th127</a></span>;<br/><span class="kw">hence </span><a NAME="E4:133_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="#E2:133"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="ref" href="semi_af1.html#T126" target="_self" title="SEMI_AF1:th.126">Th126</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="E86">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:133_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="#E1:133"><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="E2:133_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#T37" target="_self" title="SEMI_AF1:th.37">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="E4:133_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#T62" target="_self" title="SEMI_AF1:th.62">Th62</a></span>;<br/><a NAME="E5:133_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="#E1:133"><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="E6:133_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="E7:133_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#T17" target="_self" title="SEMI_AF1:th.17">Th17</a></span>;<br/><a NAME="E8:133_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#T12" target="_self" title="SEMI_AF1:th.12">Th12</a></span>;<br/><a NAME="E9:133_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="#E4:133_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#T54" target="_self" title="SEMI_AF1:th.54">Th54</a></span>;<br/><a NAME="E10:133_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="#E7:133_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="E11:133_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="#E4:133_2"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="ref" href="semi_af1.html#T56" target="_self" title="SEMI_AF1:th.56">Th56</a></span>;<br/><a NAME="E12:133_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="#E3:133"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="ref" href="semi_af1.html#T118" target="_self" title="SEMI_AF1:th.118">Th118</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E13:133_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="E14:133_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="E15:133_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="Red">::  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="E16:133_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="#E1:133"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E13:133_2"><span class="lab"><font color="Green" title="E93">A13</font></span></a>, <a class="ref" href="semi_af1.html#T123" target="_self" title="SEMI_AF1:th.123">Th123</a></span>;<br/><a NAME="E17:133_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#T37" target="_self" title="SEMI_AF1:th.37">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="E19:133_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#T62" target="_self" title="SEMI_AF1:th.62">Th62</a></span>;<br/><a NAME="E20:133_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="#E4:133_2"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="ref" href="semi_af1.html#T56" target="_self" title="SEMI_AF1:th.56">Th56</a></span>;<br/><a NAME="E21:133_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="#E2:133"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="ref" href="semi_af1.html#T118" target="_self" title="SEMI_AF1:th.118">Th118</a></span>;<br/><a NAME="E22:133_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="#E1:133"><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="E23:133_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#T17" target="_self" title="SEMI_AF1:th.17">Th17</a></span>;<br/><a NAME="E24:133_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="#E1:133"><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="E25:133_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="#E4:133_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#T54" target="_self" title="SEMI_AF1:th.54">Th54</a></span>;<br/><a NAME="E26:133_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="#E23:133_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="E27:133_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="#E4:133_2"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="ref" href="semi_af1.html#T54" target="_self" title="SEMI_AF1:th.54">Th54</a></span>;<br/><a NAME="E28:133_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="#E19:133_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#T54" target="_self" title="SEMI_AF1:th.54">Th54</a></span>;<br/><a NAME="E29:133_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="#E26:133_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="E30:133_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="E31:133_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="#E19:133_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#T54" target="_self" title="SEMI_AF1:th.54">Th54</a></span>;<br/><a NAME="E32:133_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="#E10:133_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="E33:133_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="#E4:133_2"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="ref" href="semi_af1.html#T56" target="_self" title="SEMI_AF1:th.56">Th56</a></span>;<br/><a NAME="E34:133_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="#E30:133_2"><span class="lab"><font color="Green" title="E103">A23</font></span></a>, <a class="txt" href="#E32:133_2"><span class="lab"><font color="Green" title="E104">A24</font></span></a>, <a class="txt" href="#E24:133_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="E35:133_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="#E4:133_2"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="ref" href="semi_af1.html#T56" target="_self" title="SEMI_AF1:th.56">Th56</a></span>;<br/><a NAME="E36:133_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="#E1:133"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E34:133_2"><span class="lab"><font color="Green" title="E105">A25</font></span></a>, <a class="ref" href="semi_af1.html#T127" target="_self" title="SEMI_AF1:th.127">Th127</a></span>;<br/><a NAME="E37:133_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="#E15:133_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="E38:133_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="#E3:133"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="txt" href="#E36:133_2"><span class="lab"><font color="Green" title="E106">A26</font></span></a>, <a class="txt" href="#E27:133_2"><span class="lab"><font color="Green" title="E102">A22</font></span></a>, <a class="txt" href="#E12:133_2"><span class="lab"><font color="Green" title="E92">A12</font></span></a>, <a class="txt" href="#E20:133_2"><span class="lab"><font color="Green" title="E97">A17</font></span></a>, <a class="txt" href="#E8:133_2"><span class="lab"><font color="Green" title="E89">A9</font></span></a>, <a class="ref" href="semi_af1.html#T39" target="_self" title="SEMI_AF1:th.39">Th39</a>, <a class="ref" href="semi_af1.html#T127" target="_self" title="SEMI_AF1:th.127">Th127</a></span>;<br/><a NAME="E39:133_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="#E14:133_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="E40:133_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="#E2:133"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E34:133_2"><span class="lab"><font color="Green" title="E105">A25</font></span></a>, <a class="txt" href="#E11:133_2"><span class="lab"><font color="Green" title="E91">A11</font></span></a>, <a class="txt" href="#E27:133_2"><span class="lab"><font color="Green" title="E102">A22</font></span></a>, <a class="txt" href="#E21:133_2"><span class="lab"><font color="Green" title="E98">A18</font></span></a>, <a class="txt" href="#E8:133_2"><span class="lab"><font color="Green" title="E89">A9</font></span></a>, <a class="ref" href="semi_af1.html#T39" target="_self" title="SEMI_AF1:th.39">Th39</a>, <a class="ref" href="semi_af1.html#T127" target="_self" title="SEMI_AF1:th.127">Th127</a></span>;<br/><span class="kw">hence </span><a NAME="E41:133_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="#E38:133_2"><span class="lab"><font color="Green" title="E107">A27</font></span></a>, <a class="ref" href="semi_af1.html#T126" target="_self" title="SEMI_AF1:th.126">Th126</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="E87">A28</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:133_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="Red">::  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="E2:133_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="#E1:133"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="semi_af1.html#T121" target="_self" title="SEMI_AF1:th.121">Th121</a>, <a class="ref" href="semi_af1.html#T122" target="_self" title="SEMI_AF1:th.122">Th122</a></span>;<br/><a NAME="E3:133_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="#E3:133"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="ref" href="semi_af1.html#T121" target="_self" title="SEMI_AF1:th.121">Th121</a>, <a class="ref" href="semi_af1.html#T122" target="_self" title="SEMI_AF1:th.122">Th122</a></span>;<br/><a NAME="E4:133_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="#E2:133"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E1:133_3"><span class="lab"><font color="Green" title="E88">A29</font></span></a>, <a class="ref" href="semi_af1.html#T121" target="_self" title="SEMI_AF1:th.121">Th121</a>, <a class="ref" href="semi_af1.html#T122" target="_self" title="SEMI_AF1:th.122">Th122</a></span>;<br/><span class="kw">hence </span><a NAME="E5:133_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="#E3:133_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="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="133_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:133_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="Red">::  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="E2:133_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="#E1:133"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="ref" href="semi_af1.html#T127" target="_self" title="SEMI_AF1:th.127">Th127</a></span>;<br/><span class="kw">hence </span><a NAME="E3:133_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="#E3:133"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="ref" href="semi_af1.html#T126" target="_self" title="SEMI_AF1:th.126">Th126</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="E8:133"/>
<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="#E4:133"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E6:133"><span class="lab"><font color="Green" title="E87">A28</font></span></a>, <a class="txt" href="#E5:133"><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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
