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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font>  ex <font color="Olive" title="b3">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 <br/>( <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">p</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">p</font> )</span><br/><b>let </b><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">a</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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 <br/>( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">p</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">p</font> )</span><br/>





<b>consider </b><font color="Maroon" title="c4">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font><b> such that </b><br/><a NAME="E2:106"/><i><font color="Green" title="E83">A1</font></i>: 
for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font> holds <br/> ( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">p</font> &amp;  ex <font color="Olive" title="b3">d</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/>( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">b</font> implies ( <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b2">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b3">d</font> &amp; <font color="Maroon" title="c4">p</font>,<font color="Olive" title="b2">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">b</font>,<font color="Olive" title="b3">d</font> ) ) )
 <b>by </b><i><a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">p</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">p</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="c4">p</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:106_1"/>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">p</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="#E2:106"><i><font color="Green" title="E83">A1</font></i></a>, <a class="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font>  ex <font color="Olive" title="b3">d</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/>( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">p</font>,<font color="Olive" title="b1">b</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  implies ( <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b2">c</font>,<font color="Olive" title="b3">d</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p</font>,<font color="Olive" title="b2">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">b</font>,<font color="Olive" title="b3">d</font> ) )</span><br/><b>let </b><font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d</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/>( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  implies ( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">c</font>,<font color="Olive" title="b1">d</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Olive" title="b1">d</font> ) )</span><br/><b>consider </b><font color="Maroon" title="c7">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font><b> such that </b><br/><a NAME="E3:106_1"/><i><font color="Green" title="E84">A2</font></i>: 
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b</font> implies ( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">d</font> &amp; <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">d</font> ) )
 <b>by </b><i><a class="txt" href="#E2:106"><i><font color="Green" title="E83">A1</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c8">d</font> = <font color="Maroon" title="c7">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  implies ( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">d</font> ) )</span><br/><b>assume </b><a NAME="E4:106_1"/>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">d</font> )</span><br/><b>hence </b><a NAME="E5:106_1"/>
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">d</font> )
 <b>by </b><i><a class="txt" href="#E3:106_1"><i><font color="Green" title="E84">A2</font></i></a>, <a class="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:106"/>
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">p</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="c4">p</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#D9" target="_self" title="SEMI_AF1:def.9">Def9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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