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<span class="kw">let </span><font color="Maroon" title="c1">PCPP</font> be   <a href="anproj_2.html#V6" title="ANPROJ_2:attr.6">Pappian</a>  <a href="anproj_2.html#NM2" title="ANPROJ_2:NM.2">CollProjectivePlane</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">b1</font>, <font color="Olive" title="b3">a1</font>, <font color="Olive" title="b4">b2</font>, <font color="Olive" title="b5">a2</font>, <font color="Olive" title="b6">b3</font>, <font color="Olive" title="b7">a3</font>, <font color="Olive" title="b8">c1</font>, <font color="Olive" title="b9">c3</font>, <font color="Olive" title="b10">c2</font>, <font color="Olive" title="b11">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PCPP</font>  st <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Olive" title="b3">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">b2</font> &amp; <font color="Olive" title="b5">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">b2</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">b3</font> &amp; <font color="Olive" title="b7">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">b3</font> &amp; not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">a1</font>,<font color="Olive" title="b5">a2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">a1</font>,<font color="Olive" title="b7">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b5">a2</font>,<font color="Olive" title="b7">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b8">c1</font>,<font color="Olive" title="b9">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">a1</font>,<font color="Olive" title="b5">a2</font>,<font color="Olive" title="b9">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">b1</font>,<font color="Olive" title="b4">b2</font>,<font color="Olive" title="b9">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b5">a2</font>,<font color="Olive" title="b7">a3</font>,<font color="Olive" title="b8">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b4">b2</font>,<font color="Olive" title="b6">b3</font>,<font color="Olive" title="b8">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">a1</font>,<font color="Olive" title="b7">a3</font>,<font color="Olive" title="b10">c2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">b1</font>,<font color="Olive" title="b6">b3</font>,<font color="Olive" title="b10">c2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">a1</font>,<font color="Olive" title="b2">b1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b5">a2</font>,<font color="Olive" title="b4">b2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b7">a3</font>,<font color="Olive" title="b6">b3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b5">a2</font>,<font color="Olive" title="b11">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">a1</font>,<font color="Olive" title="b7">a3</font>,<font color="Olive" title="b11">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b8">c1</font>,<font color="Olive" title="b9">c3</font>,<font color="Olive" title="b11">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/><font color="Olive" title="b8">c1</font>,<font color="Olive" title="b10">c2</font>,<font color="Olive" title="b9">c3</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">o</font>, <font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c4">a1</font>, <font color="Maroon" title="c5">b2</font>, <font color="Maroon" title="c6">a2</font>, <font color="Maroon" title="c7">b3</font>, <font color="Maroon" title="c8">a3</font>, <font color="Maroon" title="c9">c1</font>, <font color="Maroon" title="c10">c3</font>, <font color="Maroon" title="c11">c2</font>, <font color="Maroon" title="c12">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PCPP</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">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b1</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b1</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b3</font> &amp; <font color="Maroon" title="c8">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b3</font> &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c6">a2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c9">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c9">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c11">c2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c11">c2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c3">b1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c5">b2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c7">b3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c12">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c12">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c12">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  implies <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</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="E1:22"/><span class="lab"><font color="Green" title="E16">A1</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">b1</font>
 <span class="kw">and </span><br/><a NAME="E2:22"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b1</font>
 <span class="kw">and </span><br/><a NAME="E3:22"/><span class="lab"><font color="Green" title="E18">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="c5">b2</font>
 <span class="kw">and </span><br/><a NAME="E4:22"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font>
 <span class="kw">and </span><br/><a NAME="E5:22"/><span class="lab"><font color="Green" title="E20">A5</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="c7">b3</font>
 <span class="kw">and </span><br/><a NAME="E6:22"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c8">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b3</font>
 <span class="kw">and </span><br/><a NAME="E7:22"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c6">a2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E8:22"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E9:22"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E10:22"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E11:22"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E12:22"/><span class="lab"><font color="Green" title="E27">A12</font></span>: 
<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E13:22"/><span class="lab"><font color="Green" title="E28">A13</font></span>: 
<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c9">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E14:22"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c9">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E15:22"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c11">c2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E16:22"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c11">c2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E17:22"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c3">b1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E18:22"/><span class="lab"><font color="Green" title="E33">A18</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c5">b2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E19:22"/><span class="lab"><font color="Green" title="E34">A19</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c7">b3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E20:22"/><span class="lab"><font color="Green" title="E35">A20</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c12">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E21:22"/><span class="lab"><font color="Green" title="E36">A21</font></span>: 
<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c12">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E22:22"/><span class="lab"><font color="Green" title="E37">A22</font></span>: 
not <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c12">x</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="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/>

<a NAME="E23:22"/><span class="lab"><font color="Green" title="E38">A23</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="c4">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/>
<a NAME="E24:22"/><span class="lab"><font color="Green" title="E39">A24</font></span>: 
<font color="Maroon" title="c3">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E8:22"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E17:22"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E19:22"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></span>;<br/>
<a NAME="E25:22"/><span class="lab"><font color="Green" title="E40">A25</font></span>: 
<font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/>
<a NAME="E26:22"/><span class="lab"><font color="Green" title="E41">A26</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="c6">a2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/>
<a NAME="E27:22"/><span class="lab"><font color="Green" title="E42">A27</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="c8">a3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/>
<a NAME="E28:22"/><span class="lab"><font color="Green" title="E43">A28</font></span>: 
<font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/>
<a NAME="E29:22"/><span class="lab"><font color="Green" title="E44">A29</font></span>: 
<font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/>
<a NAME="E30:22"/>
( not <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">b1</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="#E8:22"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E17:22"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<a NAME="E31:22"/><span class="kw">then </span>
not <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c8">a3</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="#E2:22"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/>
<a NAME="E32:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A30</font></span>: 
not <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c3">b1</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E46">A31</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:22_1"/>
( not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c4">a1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c10">c3</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="#E7:22"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E12:22"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E2:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A32</font></span>: 
<font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E17:22"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="hessenbe.html#T10" target="_self" title="HESSENBE:th.10">Th10</a></span>;<br/><a NAME="E3:22_1"/>
not <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c2">o</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="#E7:22"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E4:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c2">o</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="#E11:22"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E2:22_1"><span class="lab"><font color="Green" title="E47">A32</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E5:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A33</font></span>: 
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c10">c3</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E6:22_1"/><span class="lab"><font color="Green" title="E49">A34</font></span>: 
<font color="Maroon" title="c9">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/><a NAME="E7:22_1"/>
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E21:22"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E8:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c8">a3</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="#E9:22"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E20:22"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E9:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A35</font></span>: 
not <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c2">o</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E10:22_1"/><span class="lab"><font color="Green" title="E51">A36</font></span>: 
<font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/><a NAME="E11:22_1"/>
( not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c4">a1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c10">c3</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="#E7:22"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E12:22"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E12:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A37</font></span>: 
<font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E17:22"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="hessenbe.html#T10" target="_self" title="HESSENBE:th.10">Th10</a></span>;<br/><a NAME="E13:22_1"/><span class="lab"><font color="Green" title="E53">A38</font></span>: 
<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c10">c3</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="#E11:22"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E14:22_1"/><span class="lab"><font color="Green" title="E54">A39</font></span>: 
<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c10">c3</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="#E11:22"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E15:22_1"/><span class="lab"><font color="Green" title="E55">A40</font></span>: 
<font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E12:22"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E17:22"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="hessenbe.html#T10" target="_self" title="HESSENBE:th.10">Th10</a></span>;<br/><a NAME="E16:22_1"/>
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c8">a3</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="#E3:22"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E9:22"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E17:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A41</font></span>: 
not <font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b2</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E18:22_1"/><span class="lab"><font color="Green" title="E57">A42</font></span>: 
<font color="Maroon" title="c9">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/><a NAME="E19:22_1"/><span class="lab"><font color="Green" title="E58">A43</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">a1</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="#E17:22"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E20:22_1"/><span class="lab"><font color="Green" title="E59">A44</font></span>: 
not <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c2">o</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="#E8:22"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E21:22_1"/>
( not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c6">a2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c9">c1</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="#E9:22"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E14:22"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E22:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A45</font></span>: 
<font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E19:22"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="hessenbe.html#T10" target="_self" title="HESSENBE:th.10">Th10</a></span>;<br/><a NAME="E23:22_1"/>
<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c2">o</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="#E19:22"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E24:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">b3</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="#E3:22"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E9:22"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="hessenbe.html#T16" target="_self" title="HESSENBE:th.16">Th16</a></span>;<br/><a NAME="E25:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A46</font></span>: 
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c5">b2</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E26:22_1"/><span class="lab"><font color="Green" title="E62">A47</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="c6">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E27:22_1"/><span class="lab"><font color="Green" title="E63">A48</font></span>: 
not <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E28:22_1"/><span class="lab"><font color="Green" title="E64">A49</font></span>: 
not <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c7">b3</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="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><a NAME="E29:22_1"/><span class="lab"><font color="Green" title="E65">A50</font></span>: 
<font color="Maroon" title="c3">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:22_1"><span class="lab"><font color="Green" title="E64">A49</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PCPP</font><span class="kw"> such that </span><br/><a NAME="E31:22_1"/><span class="lab"><font color="Green" title="E66">A51</font></span>: 
<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c13">z</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E32:22_1"/><span class="lab"><font color="Green" title="E67">A52</font></span>: 
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c13">z</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="anproj_2.html#D14" title="ANPROJ_2:def.14">ANPROJ_2:def 14</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PCPP</font><span class="kw"> such that </span><br/><a NAME="E34:22_1"/><span class="lab"><font color="Green" title="E68">A53</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c14">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E35:22_1"/><span class="lab"><font color="Green" title="E69">A54</font></span>: 
<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c14">p</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="anproj_2.html#D14" title="ANPROJ_2:def.14">ANPROJ_2:def 14</a></span>;<br/><a NAME="E36:22_1"/><span class="lab"><font color="Green" title="E70">A55</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="c14">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E35:22_1"><span class="lab"><font color="Green" title="E69">A54</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E37:22_1"/><span class="lab"><font color="Green" title="E71">A56</font></span>: 
<font color="Maroon" title="c8">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E6:22"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E9:22"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E14:22"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E19:22"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="hessenbe.html#T10" target="_self" title="HESSENBE:th.10">Th10</a></span>;<br/><a NAME="E38:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A57</font></span>: 
<font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E27:22_1"><span class="lab"><font color="Green" title="E63">A48</font></span></a>, <a class="txt" href="#E32:22_1"><span class="lab"><font color="Green" title="E67">A52</font></span></a>, <a class="txt" href="#E15:22_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E22:22_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E12:22_1"><span class="lab"><font color="Green" title="E52">A37</font></span></a>, <a class="ref" href="hessenbe.html#T12" target="_self" title="HESSENBE:th.12">Th12</a></span>;<br/><a NAME="E39:22_1"/><span class="lab"><font color="Green" title="E73">A58</font></span>: 
<font color="Maroon" title="c8">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E27:22_1"><span class="lab"><font color="Green" title="E63">A48</font></span></a>, <a class="txt" href="#E32:22_1"><span class="lab"><font color="Green" title="E67">A52</font></span></a>, <a class="txt" href="#E15:22_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E22:22_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E12:22_1"><span class="lab"><font color="Green" title="E52">A37</font></span></a>, <a class="txt" href="#E37:22_1"><span class="lab"><font color="Green" title="E71">A56</font></span></a>, <a class="ref" href="hessenbe.html#T12" target="_self" title="HESSENBE:th.12">Th12</a></span>;<br/><a NAME="E40:22_1"/><span class="lab"><font color="Green" title="E74">A59</font></span>: 
<font color="Maroon" title="c8">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:22_1_1"/>
not <font color="Maroon" title="c8">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A60</font></span>: 
<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c13">z</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="#E34:22_1"><span class="lab"><font color="Green" title="E68">A53</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<a NAME="E3:22_1_1"/>
( <font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c13">z</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a1</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="#E8:22"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E31:22_1"><span class="lab"><font color="Green" title="E66">A51</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:22_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:22_1"><span class="lab"><font color="Green" title="E73">A58</font></span></a>, <a class="txt" href="#E2:22_1_1"><span class="lab"><font color="Green" title="E75">A60</font></span></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</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><a NAME="E41:22_1"/><span class="lab"><font color="Green" title="E75">A61</font></span>: 
<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c12">x</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="#E21:22"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E42:22_1"/><span class="lab"><font color="Green" title="E76">A62</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c6">a2</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="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E43:22_1"/>
<font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E27:22_1"><span class="lab"><font color="Green" title="E63">A48</font></span></a>, <a class="txt" href="#E32:22_1"><span class="lab"><font color="Green" title="E67">A52</font></span></a>, <a class="txt" href="#E15:22_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E22:22_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E12:22_1"><span class="lab"><font color="Green" title="E52">A37</font></span></a>, <a class="txt" href="#E37:22_1"><span class="lab"><font color="Green" title="E71">A56</font></span></a>, <a class="ref" href="hessenbe.html#T12" target="_self" title="HESSENBE:th.12">Th12</a></span>;<br/><a NAME="E44:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c2">o</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="#E31:22_1"><span class="lab"><font color="Green" title="E66">A51</font></span></a>, <a class="txt" href="#E20:22_1"><span class="lab"><font color="Green" title="E59">A44</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E45:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c4">a1</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E46:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c4">a1</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="#E34:22_1"><span class="lab"><font color="Green" title="E68">A53</font></span></a>, <a class="txt" href="#E36:22_1"><span class="lab"><font color="Green" title="E70">A55</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E47:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A63</font></span>: 
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c14">p</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E48:22_1"/><span class="lab"><font color="Green" title="E78">A64</font></span>: 
<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c4">a1</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="#E31:22_1"><span class="lab"><font color="Green" title="E66">A51</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E49:22_1"/><span class="lab"><font color="Green" title="E79">A65</font></span>: 
<font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</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="#E35:22_1"><span class="lab"><font color="Green" title="E69">A54</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PCPP</font><span class="kw"> such that </span><br/><a NAME="E51:22_1"/><span class="lab"><font color="Green" title="E80">A66</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c15">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E52:22_1"/><span class="lab"><font color="Green" title="E81">A67</font></span>: 
<font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c15">q</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="anproj_2.html#D14" title="ANPROJ_2:def.14">ANPROJ_2:def 14</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PCPP</font><span class="kw"> such that </span><br/><a NAME="E54:22_1"/><span class="lab"><font color="Green" title="E82">A68</font></span>: 
<font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c16">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E55:22_1"/><span class="lab"><font color="Green" title="E83">A69</font></span>: 
<font color="Maroon" title="c15">q</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c16">r</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="anproj_2.html#D14" title="ANPROJ_2:def.14">ANPROJ_2:def 14</a></span>;<br/><a NAME="E56:22_1"/><span class="lab"><font color="Green" title="E84">A70</font></span>: 
<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c15">q</font>,<font color="Maroon" title="c16">r</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="#E55:22_1"><span class="lab"><font color="Green" title="E83">A69</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E57:22_1"/>
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c9">c1</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="#E13:22"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E19:22"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E58:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A71</font></span>: 
<font color="Maroon" title="c5">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E27:22"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="txt" href="#E42:22_1"><span class="lab"><font color="Green" title="E76">A62</font></span></a>, <a class="txt" href="#E25:22_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="ref" href="hessenbe.html#T10" target="_self" title="HESSENBE:th.10">Th10</a></span>;<br/><a NAME="E59:22_1"/><span class="lab"><font color="Green" title="E86">A72</font></span>: 
<font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E12:22"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E17:22"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="hessenbe.html#T10" target="_self" title="HESSENBE:th.10">Th10</a></span>;<br/><a NAME="E60:22_1"/>
not <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c2">o</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="#E8:22"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E61:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c2">o</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="#E31:22_1"><span class="lab"><font color="Green" title="E66">A51</font></span></a>, <a class="txt" href="#E38:22_1"><span class="lab"><font color="Green" title="E72">A57</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E62:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c4">a1</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E63:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c4">a1</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="#E34:22_1"><span class="lab"><font color="Green" title="E68">A53</font></span></a>, <a class="txt" href="#E36:22_1"><span class="lab"><font color="Green" title="E70">A55</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E64:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A73</font></span>: 
<font color="Maroon" title="c3">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:22"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PCPP</font><span class="kw"> such that </span><br/><a NAME="E66:22_1"/><span class="lab"><font color="Green" title="E88">A74</font></span>: 
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c17">a</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E67:22_1"/><span class="lab"><font color="Green" title="E89">A75</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c17">a</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="anproj_2.html#D14" title="ANPROJ_2:def.14">ANPROJ_2:def 14</a></span>;<br/><a NAME="E68:22_1"/><span class="lab"><font color="Green" title="E90">A76</font></span>: 
<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c17">a</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="#E66:22_1"><span class="lab"><font color="Green" title="E88">A74</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E69:22_1"/><span class="lab"><font color="Green" title="E91">A77</font></span>: 
<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</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="#E67:22_1"><span class="lab"><font color="Green" title="E89">A75</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E70:22_1"/><span class="lab"><font color="Green" title="E92">A78</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="c17">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E66:22_1"><span class="lab"><font color="Green" title="E88">A74</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E71:22_1"/><span class="lab"><font color="Green" title="E93">A79</font></span>: 
<font color="Maroon" title="c9">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/><a NAME="E72:22_1"/><span class="lab"><font color="Green" title="E94">A80</font></span>: 
<font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:22_1_2"/><span class="lab"><font color="Green" title="E95">A81</font></span>: 
<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c4">a1</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="#E11:22"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:22_1_2"/><span class="lab"><font color="Green" title="E96">A82</font></span>: 
not <font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:22_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c9">c1</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="#E66:22_1"><span class="lab"><font color="Green" title="E88">A74</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<a NAME="E4:22_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c4">a1</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="#E15:22_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E1:22_1_2"><span class="lab"><font color="Green" title="E95">A81</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/>
<a NAME="E5:22_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A83</font></span>: 
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c4">a1</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<a NAME="E6:22_1_2"/><span class="lab"><font color="Green" title="E98">A84</font></span>: 
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</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="#E13:22"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<a NAME="E7:22_1_2"/>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c10">c3</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="#E66:22_1"><span class="lab"><font color="Green" title="E88">A74</font></span></a>, <a class="txt" href="#E2:22_1_2"><span class="lab"><font color="Green" title="E96">A82</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<a NAME="E8:22_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c8">a3</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="#E22:22_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E6:22_1_2"><span class="lab"><font color="Green" title="E98">A84</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:22_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:22_1"><span class="lab"><font color="Green" title="E63">A48</font></span></a>, <a class="txt" href="#E66:22_1"><span class="lab"><font color="Green" title="E88">A74</font></span></a>, <a class="txt" href="#E71:22_1"><span class="lab"><font color="Green" title="E93">A79</font></span></a>, <a class="txt" href="#E2:22_1_2"><span class="lab"><font color="Green" title="E96">A82</font></span></a>, <a class="txt" href="#E5:22_1_2"><span class="lab"><font color="Green" title="E97">A83</font></span></a>, <a class="ref" href="anproj_2.html#D8" title="ANPROJ_2:def.8">ANPROJ_2:def 8</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="E73:22_1"/><span class="lab"><font color="Green" title="E95">A85</font></span>: 
not <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</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"> contradiction</span><br/><a NAME="E74:22_1"/>
<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c13">z</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="#E32:22_1"><span class="lab"><font color="Green" title="E67">A52</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E75:22_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c13">z</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="#E68:22_1"><span class="lab"><font color="Green" title="E90">A76</font></span></a>, <a class="txt" href="#E18:22_1"><span class="lab"><font color="Green" title="E57">A42</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E76:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A86</font></span>: 
<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c10">c3</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E77:22_1"/><span class="lab"><font color="Green" title="E97">A87</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c17">a</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="#E20:22"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E26:22"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="txt" href="#E67:22_1"><span class="lab"><font color="Green" title="E89">A75</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">q9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PCPP</font><span class="kw"> such that </span><br/><a NAME="E79:22_1"/><span class="lab"><font color="Green" title="E98">A88</font></span>: 
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c18">q9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c18">q9</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="anproj_2.html#D14" title="ANPROJ_2:def.14">ANPROJ_2:def 14</a></span>;<br/><a NAME="E80:22_1"/>
<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c14">p</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="#E35:22_1"><span class="lab"><font color="Green" title="E69">A54</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E81:22_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c18">q9</font>,<font color="Maroon" title="c14">p</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="#E9:22"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E34:22_1"><span class="lab"><font color="Green" title="E68">A53</font></span></a>, <a class="txt" href="#E67:22_1"><span class="lab"><font color="Green" title="E89">A75</font></span></a>, <a class="txt" href="#E79:22_1"><span class="lab"><font color="Green" title="E98">A88</font></span></a>, <a class="txt" href="#E72:22_1"><span class="lab"><font color="Green" title="E94">A80</font></span></a>, <a class="txt" href="#E70:22_1"><span class="lab"><font color="Green" title="E92">A78</font></span></a>, <a class="txt" href="#E10:22_1"><span class="lab"><font color="Green" title="E51">A36</font></span></a>, <a class="txt" href="#E38:22_1"><span class="lab"><font color="Green" title="E72">A57</font></span></a>, <a class="txt" href="#E39:22_1"><span class="lab"><font color="Green" title="E73">A58</font></span></a>, <a class="txt" href="#E48:22_1"><span class="lab"><font color="Green" title="E78">A64</font></span></a>, <a class="txt" href="#E14:22_1"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E76:22_1"><span class="lab"><font color="Green" title="E96">A86</font></span></a>, <a class="ref" href="hessenbe.html#T20" target="_self" title="HESSENBE:th.20">Th20</a></span>;<br/><a NAME="E82:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A89</font></span>: 
<font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c18">q9</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E83:22_1"/>
not <font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</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="#E27:22_1"><span class="lab"><font color="Green" title="E63">A48</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E84:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c8">a3</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="#E59:22_1"><span class="lab"><font color="Green" title="E86">A72</font></span></a>, <a class="txt" href="#E13:22_1"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E85:22_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:22_1"><span class="lab"><font color="Green" title="E69">A54</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E86:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A90</font></span>: 
<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c15">q</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="#E23:22"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="txt" href="#E51:22_1"><span class="lab"><font color="Green" title="E80">A66</font></span></a>, <a class="txt" href="#E52:22_1"><span class="lab"><font color="Green" title="E81">A67</font></span></a>, <a class="txt" href="#E79:22_1"><span class="lab"><font color="Green" title="E98">A88</font></span></a>, <a class="txt" href="#E82:22_1"><span class="lab"><font color="Green" title="E99">A89</font></span></a>, <a class="txt" href="#E47:22_1"><span class="lab"><font color="Green" title="E77">A63</font></span></a>, <a class="ref" href="hessenbe.html#T17" target="_self" title="HESSENBE:th.17">Th17</a></span>;<br/><a NAME="E87:22_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c15">q</font>,<font color="Maroon" title="c17">a</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E88:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A91</font></span>: 
<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c16">r</font>,<font color="Maroon" title="c17">a</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="#E8:22"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E51:22_1"><span class="lab"><font color="Green" title="E80">A66</font></span></a>, <a class="txt" href="#E56:22_1"><span class="lab"><font color="Green" title="E84">A70</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E89:22_1"/>
( not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c6">a2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c9">c1</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="#E9:22"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E14:22"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E90:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A92</font></span>: 
<font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E19:22"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="hessenbe.html#T10" target="_self" title="HESSENBE:th.10">Th10</a></span>;<br/><a NAME="E91:22_1"/><span class="lab"><font color="Green" title="E103">A93</font></span>: 
<font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:22_1_3"/><span class="lab"><font color="Green" title="E104">A94</font></span>: 
<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c4">a1</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="#E11:22"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:22_1_3"/><span class="lab"><font color="Green" title="E105">A95</font></span>: 
not <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:22_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c9">c1</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="#E66:22_1"><span class="lab"><font color="Green" title="E88">A74</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<a NAME="E4:22_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c4">a1</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="#E59:22_1"><span class="lab"><font color="Green" title="E86">A72</font></span></a>, <a class="txt" href="#E1:22_1_3"><span class="lab"><font color="Green" title="E104">A94</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/>
<a NAME="E5:22_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A96</font></span>: 
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c4">a1</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<a NAME="E6:22_1_3"/><span class="lab"><font color="Green" title="E107">A97</font></span>: 
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</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="#E13:22"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<a NAME="E7:22_1_3"/>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c10">c3</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="#E66:22_1"><span class="lab"><font color="Green" title="E88">A74</font></span></a>, <a class="txt" href="#E2:22_1_3"><span class="lab"><font color="Green" title="E105">A95</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<a NAME="E8:22_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c8">a3</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="#E90:22_1"><span class="lab"><font color="Green" title="E102">A92</font></span></a>, <a class="txt" href="#E6:22_1_3"><span class="lab"><font color="Green" title="E107">A97</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:22_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:22_1"><span class="lab"><font color="Green" title="E63">A48</font></span></a>, <a class="txt" href="#E66:22_1"><span class="lab"><font color="Green" title="E88">A74</font></span></a>, <a class="txt" href="#E6:22_1"><span class="lab"><font color="Green" title="E49">A34</font></span></a>, <a class="txt" href="#E2:22_1_3"><span class="lab"><font color="Green" title="E105">A95</font></span></a>, <a class="txt" href="#E5:22_1_3"><span class="lab"><font color="Green" title="E106">A96</font></span></a>, <a class="ref" href="anproj_2.html#D8" title="ANPROJ_2:def.8">ANPROJ_2:def 8</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">consider </span><font color="Maroon" title="c19">z99</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PCPP</font><span class="kw"> such that </span><br/><a NAME="E93:22_1"/><span class="lab"><font color="Green" title="E104">A98</font></span>: 
( <font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c16">r</font>,<font color="Maroon" title="c19">z99</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c19">z99</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="anproj_2.html#D14" title="ANPROJ_2:def.14">ANPROJ_2:def 14</a></span>;<br/><a NAME="E94:22_1"/>
<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c2">o</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="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E95:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A99</font></span>: 
not <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b2</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="#E1:22"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E17:22"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="hessenbe.html#T16" target="_self" title="HESSENBE:th.16">Th16</a></span>;<br/><a NAME="E96:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">b2</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E97:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A100</font></span>: 
<font color="Maroon" title="c5">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E23:22"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="txt" href="#E19:22_1"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="txt" href="#E42:22_1"><span class="lab"><font color="Green" title="E76">A62</font></span></a>, <a class="ref" href="hessenbe.html#T10" target="_self" title="HESSENBE:th.10">Th10</a></span>;<br/><a NAME="E98:22_1"/><span class="lab"><font color="Green" title="E107">A101</font></span>: 
<font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:22_1_4"/><span class="lab"><font color="Green" title="E108">A102</font></span>: 
<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c3">b1</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="#E12:22"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:22_1_4"/><span class="lab"><font color="Green" title="E109">A103</font></span>: 
not <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:22_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c9">c1</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="#E66:22_1"><span class="lab"><font color="Green" title="E88">A74</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<a NAME="E4:22_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c3">b1</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="#E97:22_1"><span class="lab"><font color="Green" title="E106">A100</font></span></a>, <a class="txt" href="#E1:22_1_4"><span class="lab"><font color="Green" title="E108">A102</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/>
<a NAME="E5:22_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A104</font></span>: 
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c3">b1</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<a NAME="E6:22_1_4"/><span class="lab"><font color="Green" title="E111">A105</font></span>: 
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c7">b3</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="#E14:22"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<a NAME="E7:22_1_4"/>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c10">c3</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="#E66:22_1"><span class="lab"><font color="Green" title="E88">A74</font></span></a>, <a class="txt" href="#E2:22_1_4"><span class="lab"><font color="Green" title="E109">A103</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/>
<a NAME="E8:22_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c7">b3</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="#E58:22_1"><span class="lab"><font color="Green" title="E85">A71</font></span></a>, <a class="txt" href="#E6:22_1_4"><span class="lab"><font color="Green" title="E111">A105</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:22_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:22_1"><span class="lab"><font color="Green" title="E64">A49</font></span></a>, <a class="txt" href="#E66:22_1"><span class="lab"><font color="Green" title="E88">A74</font></span></a>, <a class="txt" href="#E6:22_1"><span class="lab"><font color="Green" title="E49">A34</font></span></a>, <a class="txt" href="#E2:22_1_4"><span class="lab"><font color="Green" title="E109">A103</font></span></a>, <a class="txt" href="#E5:22_1_4"><span class="lab"><font color="Green" title="E110">A104</font></span></a>, <a class="ref" href="anproj_2.html#D8" title="ANPROJ_2:def.8">ANPROJ_2:def 8</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><a NAME="E99:22_1"/>
not <font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c2">o</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="#E9:22"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E100:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c2">o</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="#E13:22"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E90:22_1"><span class="lab"><font color="Green" title="E102">A92</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E101:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A106</font></span>: 
<font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:22_1"><span class="lab"><font color="Green" title="E89">A75</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E102:22_1"/>
( not <font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c2">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c10">c3</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="#E7:22"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E103:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c2">o</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="#E59:22_1"><span class="lab"><font color="Green" title="E86">A72</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E104:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A107</font></span>: 
<font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:22_1"><span class="lab"><font color="Green" title="E89">A75</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E105:22_1"/>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c5">b2</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="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E26:22"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="txt" href="#E67:22_1"><span class="lab"><font color="Green" title="E89">A75</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E106:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A108</font></span>: 
<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b2</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E107:22_1"/>
( <font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">a3</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="#E9:22"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E108:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A109</font></span>: 
not <font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c8">a3</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="#E4:22"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E109:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A110</font></span>: 
<font color="Maroon" title="c5">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:22_1"><span class="lab"><font color="Green" title="E69">A54</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E110:22_1"/>
<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c13">z</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="#E31:22_1"><span class="lab"><font color="Green" title="E66">A51</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E111:22_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">z</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="#E10:22_1"><span class="lab"><font color="Green" title="E51">A36</font></span></a>, <a class="txt" href="#E41:22_1"><span class="lab"><font color="Green" title="E75">A61</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E112:22_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c13">z</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E113:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c2">o</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="#E22:22"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E32:22_1"><span class="lab"><font color="Green" title="E67">A52</font></span></a>, <a class="txt" href="#E9:22_1"><span class="lab"><font color="Green" title="E50">A35</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E114:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c12">x</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E115:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c12">x</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="#E34:22_1"><span class="lab"><font color="Green" title="E68">A53</font></span></a>, <a class="txt" href="#E36:22_1"><span class="lab"><font color="Green" title="E70">A55</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E116:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">p</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E117:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A111</font></span>: 
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c14">p</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="#E70:22_1"><span class="lab"><font color="Green" title="E92">A78</font></span></a>, <a class="txt" href="#E77:22_1"><span class="lab"><font color="Green" title="E97">A87</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E118:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c14">p</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E119:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c14">p</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="#E91:22_1"><span class="lab"><font color="Green" title="E103">A93</font></span></a>, <a class="txt" href="#E69:22_1"><span class="lab"><font color="Green" title="E91">A77</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E120:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c17">a</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E121:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A112</font></span>: 
not <font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c17">a</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="#E40:22_1"><span class="lab"><font color="Green" title="E74">A59</font></span></a>, <a class="txt" href="#E49:22_1"><span class="lab"><font color="Green" title="E79">A65</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E122:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A113</font></span>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E88:22_1"><span class="lab"><font color="Green" title="E101">A91</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E123:22_1"/>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c5">b2</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="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E26:22"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="txt" href="#E67:22_1"><span class="lab"><font color="Green" title="E89">A75</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E124:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c14">p</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="#E3:22"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E117:22_1"><span class="lab"><font color="Green" title="E113">A111</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E125:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c2">o</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E126:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c16">r</font>,<font color="Maroon" title="c2">o</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="#E54:22_1"><span class="lab"><font color="Green" title="E82">A68</font></span></a>, <a class="txt" href="#E122:22_1"><span class="lab"><font color="Green" title="E115">A113</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E127:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A114</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:22_1"><span class="lab"><font color="Green" title="E68">A53</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c20">z9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PCPP</font><span class="kw"> such that </span><br/><a NAME="E129:22_1"/><span class="lab"><font color="Green" title="E117">A115</font></span>: 
( <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c16">r</font>,<font color="Maroon" title="c20">z9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c20">z9</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="anproj_2.html#D14" title="ANPROJ_2:def.14">ANPROJ_2:def 14</a></span>;<br/><a NAME="E130:22_1"/><span class="lab"><font color="Green" title="E118">A116</font></span>: 
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c8">a3</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="#E9:22"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E67:22_1"><span class="lab"><font color="Green" title="E89">A75</font></span></a>, <a class="txt" href="#E70:22_1"><span class="lab"><font color="Green" title="E92">A78</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E131:22_1"/><span class="kw">then </span>
not <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">a3</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E132:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A117</font></span>: 
not <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c8">a3</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="#E98:22_1"><span class="lab"><font color="Green" title="E107">A101</font></span></a>, <a class="txt" href="#E106:22_1"><span class="lab"><font color="Green" title="E110">A108</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E133:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A118</font></span>: 
<font color="Maroon" title="c5">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E88:22_1"><span class="lab"><font color="Green" title="E101">A91</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><div><span class="lab"><font color="Green" title="E121">A119</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_5"><span class="kw">now </span></a><div class="add"><a NAME="E1:22_1_5"/>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c15">q</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="#E17:22"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E23:22"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="txt" href="#E51:22_1"><span class="lab"><font color="Green" title="E80">A66</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E2:22_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A120</font></span>: 
<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c15">q</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><span class="kw">assume </span><a NAME="E3:22_1_5"/>
<font color="Maroon" title="c3">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:22_1_5"/><span class="kw">then </span>
not <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c15">q</font>,<font color="Maroon" title="c5">b2</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="#E95:22_1"><span class="lab"><font color="Green" title="E105">A99</font></span></a>, <a class="txt" href="#E2:22_1_5"><span class="lab"><font color="Green" title="E122">A120</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E5:22_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A121</font></span>: 
not <font color="Maroon" title="c15">q</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">b2</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E6:22_1_5"/><span class="lab"><font color="Green" title="E124">A122</font></span>: 
( <font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c16">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c15">q</font>,<font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c10">c3</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="#E52:22_1"><span class="lab"><font color="Green" title="E81">A67</font></span></a>, <a class="txt" href="#E54:22_1"><span class="lab"><font color="Green" title="E82">A68</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E7:22_1_5"/>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c15">q</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="#E17:22"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E23:22"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="txt" href="#E51:22_1"><span class="lab"><font color="Green" title="E80">A66</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E8:22_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A123</font></span>: 
<font color="Maroon" title="c15">q</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c2">o</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E9:22_1_5"/><span class="lab"><font color="Green" title="E126">A124</font></span>: 
( <font color="Maroon" title="c15">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">o</font> &amp; <font color="Maroon" title="c5">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:22_1"><span class="lab"><font color="Green" title="E69">A54</font></span></a>, <a class="txt" href="#E130:22_1"><span class="lab"><font color="Green" title="E118">A116</font></span></a>, <a class="txt" href="#E108:22_1"><span class="lab"><font color="Green" title="E111">A109</font></span></a>, <a class="txt" href="#E86:22_1"><span class="lab"><font color="Green" title="E100">A90</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E10:22_1_5"/>
( <font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c13">z</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c17">a</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="#E32:22_1"><span class="lab"><font color="Green" title="E67">A52</font></span></a>, <a class="txt" href="#E66:22_1"><span class="lab"><font color="Green" title="E88">A74</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E11:22_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c17">a</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="#E6:22_1"><span class="lab"><font color="Green" title="E49">A34</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E12:22_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A125</font></span>: 
<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c13">z</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E13:22_1_5"/><span class="lab"><font color="Green" title="E128">A126</font></span>: 
( not <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c2">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c13">z</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="#E10:22"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E34:22_1"><span class="lab"><font color="Green" title="E68">A53</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E14:22_1_5"/>
( <font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c9">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c14">p</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="#E13:22"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E35:22_1"><span class="lab"><font color="Green" title="E69">A54</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E15:22_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A127</font></span>: 
<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c9">c1</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="#E29:22"><span class="lab"><font color="Green" title="E44">A29</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E16:22_1_5"/>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c5">b2</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="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E26:22"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="txt" href="#E67:22_1"><span class="lab"><font color="Green" title="E89">A75</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E17:22_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A128</font></span>: 
<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c17">a</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E18:22_1_5"/>
<font color="Maroon" title="c15">q</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c17">a</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="#E86:22_1"><span class="lab"><font color="Green" title="E100">A90</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E19:22_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A129</font></span>: 
<font color="Maroon" title="c15">q</font>,<font color="Maroon" title="c16">r</font>,<font color="Maroon" title="c17">a</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="#E8:22"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E51:22_1"><span class="lab"><font color="Green" title="E80">A66</font></span></a>, <a class="txt" href="#E55:22_1"><span class="lab"><font color="Green" title="E83">A69</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E20:22_1_5"/><span class="lab"><font color="Green" title="E132">A130</font></span>: 
( <font color="Maroon" title="c5">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">r</font> &amp; <font color="Maroon" title="c14">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">r</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E132:22_1"><span class="lab"><font color="Green" title="E119">A117</font></span></a>, <a class="txt" href="#E88:22_1"><span class="lab"><font color="Green" title="E101">A91</font></span></a>, <a class="txt" href="#E121:22_1"><span class="lab"><font color="Green" title="E114">A112</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E21:22_1_5"/>
( <font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c17">a</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="#E12:22"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E26:22"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="txt" href="#E67:22_1"><span class="lab"><font color="Green" title="E89">A75</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E22:22_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c20">z9</font>,<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c10">c3</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="#E1:22"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E129:22_1"><span class="lab"><font color="Green" title="E117">A115</font></span></a>, <a class="txt" href="#E9:22_1_5"><span class="lab"><font color="Green" title="E126">A124</font></span></a>, <a class="txt" href="#E20:22_1_5"><span class="lab"><font color="Green" title="E132">A130</font></span></a>, <a class="txt" href="#E5:22_1_5"><span class="lab"><font color="Green" title="E123">A121</font></span></a>, <a class="txt" href="#E8:22_1_5"><span class="lab"><font color="Green" title="E125">A123</font></span></a>, <a class="txt" href="#E6:22_1_5"><span class="lab"><font color="Green" title="E124">A122</font></span></a>, <a class="txt" href="#E19:22_1_5"><span class="lab"><font color="Green" title="E131">A129</font></span></a>, <a class="ref" href="hessenbe.html#T20" target="_self" title="HESSENBE:th.20">Th20</a></span>;<br/><a NAME="E23:22_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A131</font></span>: 
<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c20">z9</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E24:22_1_5"/><span class="lab"><font color="Green" title="E134">A132</font></span>: 
<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c9">c1</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="#E14:22"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E25:22_1_5"/>
( <font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">b3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c16">r</font>,<font color="Maroon" title="c14">p</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="#E19:22"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E54:22_1"><span class="lab"><font color="Green" title="E82">A68</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E26:22_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c19">z99</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c17">a</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="#E5:22"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E6:22"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E93:22_1"><span class="lab"><font color="Green" title="E104">A98</font></span></a>, <a class="txt" href="#E109:22_1"><span class="lab"><font color="Green" title="E112">A110</font></span></a>, <a class="txt" href="#E88:22_1"><span class="lab"><font color="Green" title="E101">A91</font></span></a>, <a class="txt" href="#E133:22_1"><span class="lab"><font color="Green" title="E120">A118</font></span></a>, <a class="txt" href="#E17:22_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E122:22_1"><span class="lab"><font color="Green" title="E115">A113</font></span></a>, <a class="txt" href="#E17:22_1_5"><span class="lab"><font color="Green" title="E130">A128</font></span></a>, <a class="txt" href="#E15:22_1_5"><span class="lab"><font color="Green" title="E129">A127</font></span></a>, <a class="txt" href="#E24:22_1_5"><span class="lab"><font color="Green" title="E134">A132</font></span></a>, <a class="ref" href="hessenbe.html#T20" target="_self" title="HESSENBE:th.20">Th20</a></span>;<br/><a NAME="E27:22_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A133</font></span>: 
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c19">z99</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E28:22_1_5"/>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c10">c3</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="#E66:22_1"><span class="lab"><font color="Green" title="E88">A74</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E29:22_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c19">z99</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="#E101:22_1"><span class="lab"><font color="Green" title="E108">A106</font></span></a>, <a class="txt" href="#E27:22_1_5"><span class="lab"><font color="Green" title="E135">A133</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E30:22_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c16">r</font>,<font color="Maroon" title="c13">z</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="#E32:22_1"><span class="lab"><font color="Green" title="E67">A52</font></span></a>, <a class="txt" href="#E93:22_1"><span class="lab"><font color="Green" title="E104">A98</font></span></a>, <a class="txt" href="#E6:22_1"><span class="lab"><font color="Green" title="E49">A34</font></span></a>, <a class="txt" href="#E36:22_1"><span class="lab"><font color="Green" title="E70">A55</font></span></a>, <a class="txt" href="#E13:22_1_5"><span class="lab"><font color="Green" title="E128">A126</font></span></a>, <a class="ref" href="hessenbe.html#T17" target="_self" title="HESSENBE:th.17">Th17</a></span>;<br/><a NAME="E31:22_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">A134</font></span>: 
<font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c16">r</font>,<font color="Maroon" title="c7">b3</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E32:22_1_5"/>
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c13">z</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c17">a</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="#E34:22_1"><span class="lab"><font color="Green" title="E68">A53</font></span></a>, <a class="txt" href="#E117:22_1"><span class="lab"><font color="Green" title="E113">A111</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E33:22_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c16">r</font>,<font color="Maroon" title="c13">z</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="#E129:22_1"><span class="lab"><font color="Green" title="E117">A115</font></span></a>, <a class="txt" href="#E36:22_1"><span class="lab"><font color="Green" title="E70">A55</font></span></a>, <a class="txt" href="#E104:22_1"><span class="lab"><font color="Green" title="E109">A107</font></span></a>, <a class="txt" href="#E12:22_1_5"><span class="lab"><font color="Green" title="E127">A125</font></span></a>, <a class="txt" href="#E23:22_1_5"><span class="lab"><font color="Green" title="E133">A131</font></span></a>, <a class="ref" href="hessenbe.html#T17" target="_self" title="HESSENBE:th.17">Th17</a></span>;<br/><a NAME="E34:22_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c16">r</font>,<font color="Maroon" title="c3">b1</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E35:22_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c7">b3</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="#E127:22_1"><span class="lab"><font color="Green" title="E116">A114</font></span></a>, <a class="txt" href="#E31:22_1_5"><span class="lab"><font color="Green" title="E136">A134</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E36:22_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c13">z</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E37:22_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c11">c2</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="#E15:22"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E16:22"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E28:22"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="txt" href="#E32:22"><span class="lab"><font color="Green" title="E45">A30</font></span></a>, <a class="txt" href="#E29:22_1"><span class="lab"><font color="Green" title="E65">A50</font></span></a>, <a class="txt" href="#E31:22_1"><span class="lab"><font color="Green" title="E66">A51</font></span></a>, <a class="txt" href="#E32:22_1"><span class="lab"><font color="Green" title="E67">A52</font></span></a>, <a class="ref" href="hessenbe.html#T17" target="_self" title="HESSENBE:th.17">Th17</a></span>;<br/><span class="kw">hence </span><a NAME="E38:22_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E73:22_1"><span class="lab"><font color="Green" title="E95">A85</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</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><a NAME="E135:22_1"/><span class="lab"><font color="Green" title="E122">A135</font></span>: 
not <font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c17">a</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="#E117:22_1"><span class="lab"><font color="Green" title="E113">A111</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_6"><span class="kw">now </span></a><div class="add"><a NAME="E1:22_1_6"/>
( <font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c9">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c14">p</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="#E13:22"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E35:22_1"><span class="lab"><font color="Green" title="E69">A54</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E2:22_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c14">p</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="#E29:22"><span class="lab"><font color="Green" title="E44">A29</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E3:22_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A136</font></span>: 
<font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c9">c1</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E4:22_1_6"/><span class="lab"><font color="Green" title="E124">A137</font></span>: 
<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c5">b2</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="#E12:22"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E5:22_1_6"/>
( not <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">b1</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="#E8:22"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E17:22"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E6:22_1_6"/><span class="kw">then </span>
not <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c8">a3</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="#E2:22"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></span>;<br/><a NAME="E7:22_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A138</font></span>: 
not <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c3">b1</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><span class="kw">assume </span><a NAME="E8:22_1_6"/><span class="lab"><font color="Green" title="E126">A139</font></span>: 
<font color="Maroon" title="c3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E9:22_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A140</font></span>: 
<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c17">a</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="#E86:22_1"><span class="lab"><font color="Green" title="E100">A90</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E10:22_1_6"/>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c13">z</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="#E32:22_1"><span class="lab"><font color="Green" title="E67">A52</font></span></a>, <a class="txt" href="#E66:22_1"><span class="lab"><font color="Green" title="E88">A74</font></span></a>, <a class="txt" href="#E6:22_1"><span class="lab"><font color="Green" title="E49">A34</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E11:22_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A141</font></span>: 
<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c13">z</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E12:22_1_6"/>
<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c2">o</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="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E13:22_1_6"/><span class="kw">then </span>
not <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b2</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="#E1:22"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E17:22"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="hessenbe.html#T16" target="_self" title="HESSENBE:th.16">Th16</a></span>;<br/><a NAME="E14:22_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A142</font></span>: 
not <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c2">o</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">z9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PCPP</font><span class="kw"> such that </span><br/><a NAME="E16:22_1_6"/><span class="lab"><font color="Green" title="E130">A143</font></span>: 
( <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c21">z9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c21">z9</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="anproj_2.html#D14" title="ANPROJ_2:def.14">ANPROJ_2:def 14</a></span>;<br/><a NAME="E17:22_1_6"/>
<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c14">p</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="#E52:22_1"><span class="lab"><font color="Green" title="E81">A67</font></span></a>, <a class="txt" href="#E8:22_1_6"><span class="lab"><font color="Green" title="E126">A139</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E18:22_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A144</font></span>: 
<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c14">p</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="#E17:22"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E5:22_1"><span class="lab"><font color="Green" title="E48">A33</font></span></a>, <a class="txt" href="#E4:22_1_6"><span class="lab"><font color="Green" title="E124">A137</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E19:22_1_6"/>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c17">a</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="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E67:22_1"><span class="lab"><font color="Green" title="E89">A75</font></span></a>, <a class="txt" href="#E26:22_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E20:22_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c21">z9</font>,<font color="Maroon" title="c17">a</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="#E5:22"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E6:22"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E14:22"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E19:22"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E27:22"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="txt" href="#E64:22_1"><span class="lab"><font color="Green" title="E87">A73</font></span></a>, <a class="txt" href="#E109:22_1"><span class="lab"><font color="Green" title="E112">A110</font></span></a>, <a class="txt" href="#E16:22_1_6"><span class="lab"><font color="Green" title="E130">A143</font></span></a>, <a class="txt" href="#E18:22_1_6"><span class="lab"><font color="Green" title="E131">A144</font></span></a>, <a class="txt" href="#E14:22_1_6"><span class="lab"><font color="Green" title="E129">A142</font></span></a>, <a class="txt" href="#E9:22_1_6"><span class="lab"><font color="Green" title="E127">A140</font></span></a>, <a class="txt" href="#E3:22_1_6"><span class="lab"><font color="Green" title="E123">A136</font></span></a>, <a class="ref" href="hessenbe.html#T20" target="_self" title="HESSENBE:th.20">Th20</a></span>;<br/><a NAME="E21:22_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A145</font></span>: 
<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c21">z9</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E22:22_1_6"/>
<font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c13">z</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="#E34:22_1"><span class="lab"><font color="Green" title="E68">A53</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E23:22_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c13">z</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="#E36:22_1"><span class="lab"><font color="Green" title="E70">A55</font></span></a>, <a class="txt" href="#E101:22_1"><span class="lab"><font color="Green" title="E108">A106</font></span></a>, <a class="txt" href="#E135:22_1"><span class="lab"><font color="Green" title="E122">A135</font></span></a>, <a class="txt" href="#E16:22_1_6"><span class="lab"><font color="Green" title="E130">A143</font></span></a>, <a class="txt" href="#E21:22_1_6"><span class="lab"><font color="Green" title="E132">A145</font></span></a>, <a class="txt" href="#E11:22_1_6"><span class="lab"><font color="Green" title="E128">A141</font></span></a>, <a class="ref" href="hessenbe.html#T17" target="_self" title="HESSENBE:th.17">Th17</a></span>;<br/><a NAME="E24:22_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c11">c2</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="#E15:22"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E16:22"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E28:22"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="txt" href="#E29:22_1"><span class="lab"><font color="Green" title="E65">A50</font></span></a>, <a class="txt" href="#E31:22_1"><span class="lab"><font color="Green" title="E66">A51</font></span></a>, <a class="txt" href="#E32:22_1"><span class="lab"><font color="Green" title="E67">A52</font></span></a>, <a class="txt" href="#E7:22_1_6"><span class="lab"><font color="Green" title="E125">A138</font></span></a>, <a class="ref" href="hessenbe.html#T17" target="_self" title="HESSENBE:th.17">Th17</a></span>;<br/><span class="kw">hence </span><a NAME="E25:22_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E73:22_1"><span class="lab"><font color="Green" title="E95">A85</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</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">hence </span><a NAME="E137:22_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E134:22_1"><span class="lab"><font color="Green" title="E121">A119</font></span></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>
<a NAME="E34:22"/><span class="lab"><font color="Green" title="E47">A146</font></span>: 
<font color="Maroon" title="c5">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E9:22"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E19:22"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></span>;<br/>
<a NAME="E35:22"/><span class="lab"><font color="Green" title="E48">A147</font></span>: 
<font color="Maroon" title="c3">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E17:22"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E49">A148</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:22_2_1"/><span class="lab"><font color="Green" title="E50">A149</font></span>: 
<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c7">b3</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="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><a NAME="E2:22_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c3">b1</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E3:22_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c9">c1</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="#E14:22"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E34:22"><span class="lab"><font color="Green" title="E47">A146</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E4:22_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A150</font></span>: 
<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c9">c1</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E5:22_2_1"/>
<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c5">b2</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="#E1:22_2_1"><span class="lab"><font color="Green" title="E50">A149</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E6:22_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c11">c2</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="#E16:22"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E24:22"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:22_2_1"/>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</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="#E12:22"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E35:22"><span class="lab"><font color="Green" title="E48">A147</font></span></a>, <a class="txt" href="#E4:22_2_1"><span class="lab"><font color="Green" title="E51">A150</font></span></a>, <a class="ref" href="anproj_2.html#D8" title="ANPROJ_2:def.8">ANPROJ_2:def 8</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><div><span class="lab"><font color="Green" title="E50">A151</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:22_2_2"/><span class="lab"><font color="Green" title="E51">A152</font></span>: 
<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</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="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><a NAME="E2:22_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c4">a1</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E3:22_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c9">c1</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="#E13:22"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E29:22"><span class="lab"><font color="Green" title="E44">A29</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><a NAME="E4:22_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A153</font></span>: 
<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c9">c1</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="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E5:22_2_2"/>
<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c6">a2</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="#E1:22_2_2"><span class="lab"><font color="Green" title="E51">A152</font></span></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></span>;<br/><a NAME="E6:22_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c11">c2</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="#E15:22"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E28:22"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:22_2_2"/>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</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="#E11:22"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E25:22"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="txt" href="#E4:22_2_2"><span class="lab"><font color="Green" title="E52">A153</font></span></a>, <a class="ref" href="anproj_2.html#D8" title="ANPROJ_2:def.8">ANPROJ_2:def 8</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="E3:22_2"/>
( <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c7">b3</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="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><span class="kw">hence </span><a NAME="E4:22_2"/>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</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="#E2:22_2"><span class="lab"><font color="Green" title="E50">A151</font></span></a>, <a class="txt" href="#E1:22_2"><span class="lab"><font color="Green" title="E49">A148</font></span></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">hence </span><a NAME="E37:22"/>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</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="#E33:22"><span class="lab"><font color="Green" title="E46">A31</font></span></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>
