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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">o</font>, <font color="Olive" title="b2">p</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#R5" title="SEMI_AF1:pred.5">qtrap</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">p</font> holds <br/> ex <font color="Olive" title="b3">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font> st <br/>( not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">q</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp;  <a href="semi_af1.html#R5" title="SEMI_AF1:pred.5">qtrap</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">q</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">p</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#R5" title="SEMI_AF1:pred.5">qtrap</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">p</font> implies  ex <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font> st <br/>( not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">q</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp;  <a href="semi_af1.html#R5" title="SEMI_AF1:pred.5">qtrap</a> <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">q</font> ) )</span><br/>





<a NAME="E1:137"/><span class="lab"><font color="Green" title="E85">A1</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">p</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/>
<span class="kw">assume </span><a NAME="E2:137"/><span class="lab"><font color="Green" title="E86">A2</font></span>: 
 <a href="semi_af1.html#R5" title="SEMI_AF1:pred.5">qtrap</a> <font color="Maroon" title="c2">o</font>,<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">  ex <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font> st <br/>( not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">q</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp;  <a href="semi_af1.html#R5" title="SEMI_AF1:pred.5">qtrap</a> <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">q</font> )</span><br/>

<a NAME="E3:137"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A3</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T131" target="_self" title="SEMI_AF1:th.131">Th131</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">r</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="E5:137"/><span class="lab"><font color="Green" title="E88">A4</font></span>: 
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">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="#E2:137"><span class="lab"><font color="Green" title="E86">A2</font></span></a>, <a class="ref" href="semi_af1.html#T41" target="_self" title="SEMI_AF1:th.41">Th41</a>, <a class="ref" href="semi_af1.html#T131" target="_self" title="SEMI_AF1:th.131">Th131</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font><span class="kw"> such that </span><br/><a NAME="E7:137"/><span class="lab"><font color="Green" title="E89">A5</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">r</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">and </span><br/><a NAME="E8:137"/><span class="lab"><font color="Green" title="E90">A6</font></span>: 
 <a href="semi_af1.html#R5" title="SEMI_AF1:pred.5">qtrap</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T129" target="_self" title="SEMI_AF1:th.129">Th129</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">q</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c2">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>  &amp;  <a href="semi_af1.html#R5" title="SEMI_AF1:pred.5">qtrap</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">q</font> )</span><br/>

<a NAME="E9:137"/>
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">r</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:137"><span class="lab"><font color="Green" title="E89">A5</font></span></a>, <a class="txt" href="#E8:137"><span class="lab"><font color="Green" title="E90">A6</font></span></a>, <a class="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</a>, <a class="ref" href="semi_af1.html#T131" target="_self" title="SEMI_AF1:th.131">Th131</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:137"/>
( not <font color="Maroon" title="c2">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>  &amp;  <a href="semi_af1.html#R5" title="SEMI_AF1:pred.5">qtrap</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:137"><span class="lab"><font color="Green" title="E87">A3</font></span></a>, <a class="txt" href="#E5:137"><span class="lab"><font color="Green" title="E88">A4</font></span></a>, <a class="txt" href="#E8:137"><span class="lab"><font color="Green" title="E90">A6</font></span></a>, <a class="txt" href="#E1:137"><span class="lab"><font color="Green" title="E85">A1</font></span></a>, <a class="ref" href="semi_af1.html#T39" target="_self" title="SEMI_AF1:th.39">Th39</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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