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<div><b>hereby </b> <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">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font> holds <br/> ( <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b1">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b2">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  ) ) implies <font color="Maroon" title="c1">CS</font> is <a href="collsp.html#V2" title="COLLSP:attr.2">reflexive</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:63_1_1_1"/><i><font color="Green" title="E57">A1</font></i>: 
<font color="Maroon" title="c1">CS</font> is <a href="collsp.html#V2" title="COLLSP:attr.2">reflexive</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">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font> holds <br/> ( <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b1">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b2">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</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">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c3">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><a NAME="E2:63_1_1_1"/>
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="collsp.html#U1" title="COLLSP:sel.1">Collinearity</a> of <font color="Maroon" title="c1">CS</font>
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:63_1_1_1"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="collsp.html#D3" title="COLLSP:def.3">COLLSP:def 3</a></i>;<br/><b>hence </b><a NAME="E3:63_1_1_1"/>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="collsp.html#D2" title="COLLSP:def.2">COLLSP:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c3">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><a NAME="E4:63_1_1_1"/>
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="collsp.html#U1" title="COLLSP:sel.1">Collinearity</a> of <font color="Maroon" title="c1">CS</font>
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:63_1_1_1"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="collsp.html#D3" title="COLLSP:def.3">COLLSP:def 3</a></i>;<br/><b>hence </b><a NAME="E5:63_1_1_1"/>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="collsp.html#D2" title="COLLSP:def.2">COLLSP:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c3">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><a NAME="E6:63_1_1_1"/>
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c3">q</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="collsp.html#U1" title="COLLSP:sel.1">Collinearity</a> of <font color="Maroon" title="c1">CS</font>
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:63_1_1_1"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="collsp.html#D3" title="COLLSP:def.3">COLLSP:def 3</a></i>;<br/><b>hence </b><a NAME="E7:63_1_1_1"/>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c3">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="collsp.html#D2" title="COLLSP:def.2">COLLSP:def 2</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>assume </b><a NAME="E2:63_1_1"/><i><font color="Green" title="E57">A2</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font> holds <br/> ( <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b1">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b2">q</font> <a href="collsp.html#R1" title="COLLSP: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="c1">CS</font> is <a href="collsp.html#V2" title="COLLSP:attr.2">reflexive</a></span><br/>

<b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="collsp.html#D3" title="COLLSP:def.3">COLLSP:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( not <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> &amp; not <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> &amp; not <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> ) or <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="collsp.html#U1" title="COLLSP:sel.1">Collinearity</a> of <font color="Maroon" title="c1">CS</font> )</span><br/>

<b>assume </b><a NAME="E3:63_1_1"/><i><font color="Green" title="E58">A3</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> or <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> or <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="collsp.html#U1" title="COLLSP:sel.1">Collinearity</a> of <font color="Maroon" title="c1">CS</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_1_1_2"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> or <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> or <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> )
 </span><b>by </b><i><a class="txt" href="anproj_2.html#E3:63_1_1"><i><font color="Green" title="E58">A3</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_2_1"><b>suppose </b></a><a NAME="E1:63_1_1_2_1"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="collsp.html#U1" title="COLLSP:sel.1">Collinearity</a> of <font color="Maroon" title="c1">CS</font></span><br/><div class="add"><a NAME="E2:63_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:63_1_1"><i><font color="Green" title="E57">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:63_1_1_2_1"/>
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="collsp.html#U1" title="COLLSP:sel.1">Collinearity</a> of <font color="Maroon" title="c1">CS</font>
 <b>by </b><i><a class="ref" href="collsp.html#D2" title="COLLSP:def.2">COLLSP:def 2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_2_2"><b>suppose </b></a><a NAME="E1:63_1_1_2_2"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="collsp.html#U1" title="COLLSP:sel.1">Collinearity</a> of <font color="Maroon" title="c1">CS</font></span><br/><div class="add"><a NAME="E2:63_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:63_1_1"><i><font color="Green" title="E57">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:63_1_1_2_2"/>
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="collsp.html#U1" title="COLLSP:sel.1">Collinearity</a> of <font color="Maroon" title="c1">CS</font>
 <b>by </b><i><a class="ref" href="collsp.html#D2" title="COLLSP:def.2">COLLSP:def 2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_2_3"><b>suppose </b></a><a NAME="E1:63_1_1_2_3"/>
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="collsp.html#U1" title="COLLSP:sel.1">Collinearity</a> of <font color="Maroon" title="c1">CS</font></span><br/><div class="add"><a NAME="E2:63_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:63_1_1"><i><font color="Green" title="E57">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:63_1_1_2_3"/>
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="collsp.html#U1" title="COLLSP:sel.1">Collinearity</a> of <font color="Maroon" title="c1">CS</font>
 <b>by </b><i><a class="ref" href="collsp.html#D2" title="COLLSP:def.2">COLLSP:def 2</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></div><b>end;</b></div>

</div>
