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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">a1</font>, <font color="Olive" title="b2">a2</font>, <font color="Olive" title="b3">a3</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font>  st <font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font>,<font color="Olive" title="b3">a3</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  holds <br/>( <font color="Olive" title="b1">a1</font>,<font color="Olive" title="b3">a3</font>,<font color="Olive" title="b2">a2</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">a2</font>,<font color="Olive" title="b1">a1</font>,<font color="Olive" title="b3">a3</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">a2</font>,<font color="Olive" title="b3">a3</font>,<font color="Olive" title="b1">a1</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">a3</font>,<font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">a3</font>,<font color="Olive" title="b2">a2</font>,<font color="Olive" title="b1">a1</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c3">a2</font>, <font color="Maroon" title="c4">a3</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"> ( <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  implies ( <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c2">a1</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:37"/>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</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="c2">a1</font>,<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c2">a1</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  )</span><br/>

<a NAME="E2:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</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="E3:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A2</font></span>: 
( <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> &amp; <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c2">a1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T18" target="_self" title="SEMI_AF1:th.18">Th18</a></span>;<br/>
<a NAME="E4:37"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="semi_af1.html#T18" target="_self" title="SEMI_AF1:th.18">Th18</a></span>;<br/>
<a NAME="E5:37"/>
( <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font> &amp; <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="semi_af1.html#T18" target="_self" title="SEMI_AF1:th.18">Th18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:37"/>
( <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c2">a1</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</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="#E3:37"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</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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