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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> 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> holds <br/><font color="Olive" title="b3">b</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">q</font>,<font color="Olive" title="b7">r</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> 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> implies <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c8">r</font> )</span><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:131"/><span class="lab"><font color="Green" title="E81">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="E2:131"/><span class="lab"><font color="Green" title="E82">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>
 ; <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="c4">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c8">r</font></span><br/>

<a NAME="E3:131"/>
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="#E1:131"><span class="lab"><font color="Green" title="E81">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="E4:131"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A3</font></span>: 
not <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="c4">b</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="E5:131"/>
<font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">r</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="#E2:131"><span class="lab"><font color="Green" title="E82">A2</font></span></a>, <a class="ref" href="semi_af1.html#D8" target="_self" title="SEMI_AF1:def.8">Def8</a></span>;<br/>
<a NAME="E6:131"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A4</font></span>: 
<font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c8">r</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="E7:131"/>
not <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">a</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">by</span> <span class="lab"><a class="txt" href="#E2:131"><span class="lab"><font color="Green" title="E82">A2</font></span></a>, <a class="ref" href="semi_af1.html#D8" target="_self" title="SEMI_AF1:def.8">Def8</a></span>;<br/>
<a NAME="E8:131"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A5</font></span>: 
not <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="c7">c</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="E9:131"/>
<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="#E1:131"><span class="lab"><font color="Green" title="E81">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="E10:131"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A6</font></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="E11:131"/>
<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="#E1:131"><span class="lab"><font color="Green" title="E81">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="E12:131"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A7</font></span>: 
<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="c3">p</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="E13:131"/>
( <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> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131"><span class="lab"><font color="Green" title="E81">A1</font></span></a>, <a class="txt" href="#E2:131"><span class="lab"><font color="Green" title="E82">A2</font></span></a>, <a class="ref" href="semi_af1.html#D8" target="_self" title="SEMI_AF1:def.8">Def8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:131"/>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c8">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:131"><span class="lab"><font color="Green" title="E83">A3</font></span></a>, <a class="txt" href="#E12:131"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="txt" href="#E10:131"><span class="lab"><font color="Green" title="E86">A6</font></span></a>, <a class="txt" href="#E8:131"><span class="lab"><font color="Green" title="E85">A5</font></span></a>, <a class="txt" href="#E6:131"><span class="lab"><font color="Green" title="E84">A4</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/>


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