<?xml version="1.0"?>
<div class="add">

<div><span class="kw">hereby </span> <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">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font>, <font color="Olive" title="b4">r1</font>, <font color="Olive" title="b5">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</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="b4">r1</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="b5">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/><font color="Olive" title="b3">r</font>,<font color="Olive" title="b4">r1</font>,<font color="Olive" title="b5">r2</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#V3" title="COLLSP:attr.3">transitive</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:63_2_1_1"/><span class="lab"><font color="Green" title="E56">A4</font></span>: 
<font color="Maroon" title="c1">CS</font> is  <a href="collsp.html#V3" title="COLLSP:attr.3">transitive</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">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font>, <font color="Olive" title="b4">r1</font>, <font color="Olive" title="b5">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</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="b4">r1</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="b5">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/><font color="Olive" title="b3">r</font>,<font color="Olive" title="b4">r1</font>,<font color="Olive" title="b5">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">r</font>, <font color="Maroon" title="c5">r1</font>, <font color="Maroon" title="c6">r2</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font> &amp; <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>  &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">r1</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="c6">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  implies <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c6">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:63_2_1_1"/><span class="lab"><font color="Green" title="E57">A5</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">and </span><br/><a NAME="E3:63_2_1_1"/><span class="lab"><font color="Green" title="E58">A6</font></span>: 
( <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>  &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <span class="kw">and </span><br/><a NAME="E4:63_2_1_1"/><span class="lab"><font color="Green" title="E59">A7</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c6">r2</font> <a href="collsp.html#R1" title="COLLSP: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="c4">r</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c6">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><a NAME="E5:63_2_1_1"/><span class="lab"><font color="Green" title="E60">A8</font></span>: 
<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="c6">r2</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_2_1_1"><span class="lab"><font color="Green" title="E59">A7</font></span></a>, <a class="ref" href="collsp.html#D2" title="COLLSP:def.2">COLLSP:def 2</a></span>;<br/><a NAME="E6:63_2_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="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> &amp; <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="c5">r1</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_2_1_1"><span class="lab"><font color="Green" title="E58">A6</font></span></a>, <a class="ref" href="collsp.html#D2" title="COLLSP:def.2">COLLSP:def 2</a></span>;<br/><a NAME="E7:63_2_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c6">r2</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_2_1_1"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="txt" href="#E2:63_2_1_1"><span class="lab"><font color="Green" title="E57">A5</font></span></a>, <a class="txt" href="#E5:63_2_1_1"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="ref" href="collsp.html#D4" title="COLLSP:def.4">COLLSP:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E8:63_2_1_1"/>
<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c6">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="collsp.html#D2" title="COLLSP:def.2">COLLSP:def 2</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/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:63_2_1"/><span class="lab"><font color="Green" title="E56">A9</font></span>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font>, <font color="Olive" title="b4">r1</font>, <font color="Olive" title="b5">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</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="b4">r1</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="b5">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/><font color="Olive" title="b3">r</font>,<font color="Olive" title="b4">r1</font>,<font color="Olive" title="b5">r2</font> <a href="collsp.html#R1" title="COLLSP: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="c1">CS</font> is  <a href="collsp.html#V3" title="COLLSP:attr.3">transitive</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">r</font>, <font color="Maroon" title="c5">r1</font>, <font color="Maroon" title="c6">r2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CS</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="collsp.html#D4" title="COLLSP:def.4">COLLSP:def 4</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">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> or  not <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> or  not <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="c5">r1</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> or  not <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="c6">r2</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> or <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c6">r2</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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:63_2_1"/><span class="lab"><font color="Green" title="E57">A10</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">and </span><br/><a NAME="E4:63_2_1"/><span class="lab"><font color="Green" title="E58">A11</font></span>: 
( <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> &amp; <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="c5">r1</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 class="kw">and </span><br/><a NAME="E5:63_2_1"/><span class="lab"><font color="Green" title="E59">A12</font></span>: 
<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="c6">r2</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c4">r</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c6">r2</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/>

<a NAME="E6:63_2_1"/><span class="lab"><font color="Green" title="E60">A13</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c6">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_2_1"><span class="lab"><font color="Green" title="E59">A12</font></span></a>, <a class="ref" href="collsp.html#D2" title="COLLSP:def.2">COLLSP:def 2</a></span>;<br/>
<a NAME="E7:63_2_1"/>
( <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>  &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_2_1"><span class="lab"><font color="Green" title="E58">A11</font></span></a>, <a class="ref" href="collsp.html#D2" title="COLLSP:def.2">COLLSP:def 2</a></span>;<br/>
<a NAME="E8:63_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c6">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_2_1"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E3:63_2_1"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E6:63_2_1"><span class="lab"><font color="Green" title="E60">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:63_2_1"/>
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c6">r2</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 class="kw">by</span> <span class="lab"><a class="ref" href="collsp.html#D2" title="COLLSP:def.2">COLLSP:def 2</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/>


</div>
