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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p2</font>, <font color="Olive" title="b2">p3</font>, <font color="Olive" title="b3">p1</font>, <font color="Olive" title="b4">q2</font>, <font color="Olive" title="b5">q3</font>, <font color="Olive" title="b6">q1</font>, <font color="Olive" title="b7">r3</font>, <font color="Olive" title="b8">r2</font>, <font color="Olive" title="b9">r1</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">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">p3</font> &amp; <font color="Olive" title="b3">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">p3</font> &amp; <font color="Olive" title="b4">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">q3</font> &amp; <font color="Olive" title="b6">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">q2</font> &amp; <font color="Olive" title="b6">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">q3</font> &amp; not <font color="Olive" title="b3">p1</font>,<font color="Olive" title="b1">p2</font>,<font color="Olive" title="b6">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">p1</font>,<font color="Olive" title="b1">p2</font>,<font color="Olive" title="b2">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b6">q1</font>,<font color="Olive" title="b4">q2</font>,<font color="Olive" title="b5">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">q2</font>,<font color="Olive" title="b7">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b6">q1</font>,<font color="Olive" title="b1">p2</font>,<font color="Olive" title="b7">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">p1</font>,<font color="Olive" title="b5">q3</font>,<font color="Olive" title="b8">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">p3</font>,<font color="Olive" title="b6">q1</font>,<font color="Olive" title="b8">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">p2</font>,<font color="Olive" title="b5">q3</font>,<font color="Olive" title="b9">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">p3</font>,<font color="Olive" title="b4">q2</font>,<font color="Olive" title="b9">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/><font color="Olive" title="b9">r1</font>,<font color="Olive" title="b8">r2</font>,<font color="Olive" title="b7">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><b>let </b><font color="Maroon" title="c2">p2</font>, <font color="Maroon" title="c3">p3</font>, <font color="Maroon" title="c4">p1</font>, <font color="Maroon" title="c5">q2</font>, <font color="Maroon" title="c6">q3</font>, <font color="Maroon" title="c7">q1</font>, <font color="Maroon" title="c8">r3</font>, <font color="Maroon" title="c9">r2</font>, <font color="Maroon" title="c10">r1</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font> &amp; <font color="Maroon" title="c4">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q3</font> &amp; <font color="Maroon" title="c7">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q2</font> &amp; <font color="Maroon" title="c7">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q3</font> &amp; not <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  implies <font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/>



















<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font> &amp; <font color="Maroon" title="c4">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q3</font> &amp; <font color="Maroon" title="c7">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q2</font> &amp; <font color="Maroon" title="c7">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q3</font> &amp; not <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/>

<a NAME="E2:16"/><b>then </b><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font>
 
<b>by </b><i><a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></i>;<br/>
<div><i><font color="Green" title="E16">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><div><i><font color="Green" title="E18">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_1"/><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><a NAME="E2:16_1_1"/>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/><a NAME="E3:16_1_1"/><b>then </b>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_1_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/><a NAME="E4:16_1_1"/><b>then </b><i><font color="Green" title="E20">A7</font></i>: 
<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/><a NAME="E5:16_1_1"/><i><font color="Green" title="E21">A8</font></i>: 
not <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/><a NAME="E6:16_1_1"/>
<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/><a NAME="E7:16_1_1"/><b>then </b><i><font color="Green" title="E22">A9</font></i>: 
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r3</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E4:16_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="hessenbe.html#E5:16_1_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>;<br/><a NAME="E8:16_1_1"/>
<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_1_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/><a NAME="E9:16_1_1"/><b>then </b><i><font color="Green" title="E23">A10</font></i>: 
not <font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T13" target="_self" title="HESSENBE:th.13">Th13</a></i>;<br/><a NAME="E10:16_1_1"/>
<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_1_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/><a NAME="E11:16_1_1"/><b>then </b>
<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/><a NAME="E12:16_1_1"/><b>then </b><i><font color="Green" title="E24">A11</font></i>: 
<font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/><a NAME="E13:16_1_1"/>
<font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/><a NAME="E14:16_1_1"/><b>then </b>
<font color="Maroon" title="c8">r3</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E7:16_1_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="hessenbe.html#E9:16_1_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="hessenbe.html#E12:16_1_1"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>;<br/><b>hence </b><a NAME="E15:16_1_1"/>
<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><i><font color="Green" title="E19">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_2"/><i><font color="Green" title="E20">A13</font></i>: 
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><a NAME="E2:16_1_2"/><i><font color="Green" title="E21">A14</font></i>: 
<font color="Maroon" title="c10">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_1_2_1"/><i><font color="Green" title="E22">A15</font></i>: 
not <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E2:16_1_2_1"/>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_1_2"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E3:16_1_2_1"/><b>then </b>
not <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_1_2"><i><font color="Green" title="E20">A13</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_1_2_1"><i><font color="Green" title="E22">A15</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></i>;<br/>
<a NAME="E4:16_1_2_1"/><b>then </b><i><font color="Green" title="E23">A16</font></i>: 
not <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E5:16_1_2_1"/>
( <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_1_2"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E6:16_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E7:16_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E8:16_1_2_1"/>
<font color="Maroon" title="c10">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E4:16_1_2_1"><i><font color="Green" title="E23">A16</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E3:16_1_2"/>
<font color="Maroon" title="c7">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_1_2_2"/>
( <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_1_2"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E2:16_1_2_2"/><b>then </b>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_1_2"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E3:16_1_2_2"/><b>then </b><i><font color="Green" title="E22">A17</font></i>: 
<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E4:16_1_2_2"/>
not <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></i>;<br/>
<a NAME="E5:16_1_2_2"/><b>then </b><i><font color="Green" title="E23">A18</font></i>: 
not <font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E6:16_1_2_2"/>
<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E7:16_1_2_2"/>
<font color="Maroon" title="c7">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r2</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E3:16_1_2_2"><i><font color="Green" title="E22">A17</font></i></a>, <a class="txt" href="hessenbe.html#E5:16_1_2_2"><i><font color="Green" title="E23">A18</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:16_1_2"/>
<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E2:16_1_2"><i><font color="Green" title="E21">A14</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_3"/><i><font color="Green" title="E20">A19</font></i>: 
( <font color="Maroon" title="c4">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q2</font> &amp; <font color="Maroon" title="c3">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><a NAME="E2:16_1_3"/>
( <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r1</font> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r3</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_1_3_1"/>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r1</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r3</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_1_3_1_1"/>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E2:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E2:16_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E3:16_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_1_3"><i><font color="Green" title="E20">A19</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E4:16_1_3_1_1"/><b>then </b>
( <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E5:16_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E6:16_1_3_1_1"/><b>then </b><i><font color="Green" title="E21">A20</font></i>: 
<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E7:16_1_3_1_1"/>
not <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_1_3"><i><font color="Green" title="E20">A19</font></i></a>, <a class="ref" href="hessenbe.html#T14" target="_self" title="HESSENBE:th.14">Th14</a></i>;<br/>
<b>hence </b><a NAME="E8:16_1_3_1_1"/>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r1</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E6:16_1_3_1_1"><i><font color="Green" title="E21">A20</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:16_1_3_1"/>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r3</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_1_3_1_2"/>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E2:16_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_1_3"><i><font color="Green" title="E20">A19</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E3:16_1_3_1_2"/><b>then </b><i><font color="Green" title="E21">A21</font></i>: 
<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E4:16_1_3_1_2"/>
( not <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E5:16_1_3_1_2"/>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r3</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E3:16_1_3_1_2"><i><font color="Green" title="E21">A21</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E3:16_1_3"/>
<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E5:16_1"/>
<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E2:16_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="hessenbe.html#E3:16_1"><i><font color="Green" title="E19">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2"/><i><font color="Green" title="E17">A22</font></i>: 
not <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><div><i><font color="Green" title="E18">A23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_1"/><i><font color="Green" title="E19">A24</font></i>: 
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><div><i><font color="Green" title="E20">A25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_1_1"/><i><font color="Green" title="E21">A26</font></i>: 
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><a NAME="E2:16_2_1_1"/>
( <font color="Maroon" title="c7">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r3</font> &amp; <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_2_1_1_1"/>
<font color="Maroon" title="c7">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r3</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r1</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_1_1_1"/>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_1_1"><i><font color="Green" title="E21">A26</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E2:16_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_1_1"><i><font color="Green" title="E21">A26</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E3:16_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E22">A27</font></i>: 
<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E4:16_2_1_1_1_1"/>
not <font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E5:16_2_1_1_1_1"/>
<font color="Maroon" title="c7">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r3</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E3:16_2_1_1_1_1"><i><font color="Green" title="E22">A27</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:16_2_1_1_1"/>
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r1</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_1_1_2"/>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_1_1"><i><font color="Green" title="E21">A26</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E2:16_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E2:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E3:16_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E22">A28</font></i>: 
<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E4:16_2_1_1_1_2"/>
not <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_1_1"><i><font color="Green" title="E21">A26</font></i></a>, <a class="ref" href="hessenbe.html#T16" target="_self" title="HESSENBE:th.16">Th16</a></i>;<br/>
<b>hence </b><a NAME="E5:16_2_1_1_1_2"/>
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r1</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E3:16_2_1_1_1_2"><i><font color="Green" title="E22">A28</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E3:16_2_1_1"/>
<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><i><font color="Green" title="E21">A29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_1_2"/><i><font color="Green" title="E22">A30</font></i>: 
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><a NAME="E2:16_2_1_2"/>
( <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r3</font> &amp; <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_2_1_2_1"/>
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r3</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r2</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_2_1_1"/><i><font color="Green" title="E23">A31</font></i>: 
<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E2:16_2_1_2_1_1"/>
( <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_1_2"><i><font color="Green" title="E22">A30</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E3:16_2_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_1_2"><i><font color="Green" title="E22">A30</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E4:16_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E24">A32</font></i>: 
<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E5:16_2_1_2_1_1"/>
not <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_2_1_1_1"/><i><font color="Green" title="E25">A33</font></i>: 
not <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c4">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E2:16_2_1_2_1_1_1"/>
<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_1_2"><i><font color="Green" title="E22">A30</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E3:16_2_1_2_1_1_1"/><b>then </b>
not <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_1_2"><i><font color="Green" title="E22">A30</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_1_2_1_1_1"><i><font color="Green" title="E25">A33</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E4:16_2_1_2_1_1_1"/>
not <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:16_2_1_2_1_1"/>
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r3</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16_2_1_2_1_1"><i><font color="Green" title="E23">A31</font></i></a>, <a class="txt" href="hessenbe.html#E4:16_2_1_2_1_1"><i><font color="Green" title="E24">A32</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:16_2_1_2_1"/>
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r2</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_2_1_2"/>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E2:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_1_2"><i><font color="Green" title="E22">A30</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E2:16_2_1_2_1_2"/><b>then </b><i><font color="Green" title="E23">A34</font></i>: 
<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E3:16_2_1_2_1_2"/>
not <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_2_1_2_1"/>
not <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E2:16_2_1_2_1_2_1"/>
not <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:16_2_1_2_1_2"/>
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r2</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E2:16_2_1_2_1_2"><i><font color="Green" title="E23">A34</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E3:16_2_1_2"/>
<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_1_3"/><i><font color="Green" title="E22">A35</font></i>: 
( <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q3</font> &amp; <font color="Maroon" title="c4">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><a NAME="E2:16_2_1_3"/><i><font color="Green" title="E23">A36</font></i>: 
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_3_1"/>
<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c4">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16_2_1"><i><font color="Green" title="E19">A24</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E2:16_2_1_3_1"/><b>then </b>
( <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_1_3"><i><font color="Green" title="E22">A35</font></i></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></i>;<br/>
<a NAME="E3:16_2_1_3_1"/><b>then </b>
<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E2:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E4:16_2_1_3_1"/><b>then </b><i><font color="Green" title="E24">A37</font></i>: 
<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E5:16_2_1_3_1"/>
not <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_3_1_1"/><i><font color="Green" title="E25">A38</font></i>: 
not <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16_2"><i><font color="Green" title="E17">A22</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E2:16_2_1_3_1_1"/>
<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E3:16_2_1_3_1_1"/><b>then </b>
not <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_1_3_1_1"><i><font color="Green" title="E25">A38</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E4:16_2_1_3_1_1"/>
not <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:16_2_1_3_1"/>
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r1</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E4:16_2_1_3_1"><i><font color="Green" title="E24">A37</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E3:16_2_1_3"/>
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_3_2"/>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16_2_1"><i><font color="Green" title="E19">A24</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E2:16_2_1_3_2"/><b>then </b>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_1_3"><i><font color="Green" title="E22">A35</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E3:16_2_1_3_2"/><b>then </b>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E2:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E4:16_2_1_3_2"/><b>then </b><i><font color="Green" title="E24">A39</font></i>: 
<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E5:16_2_1_3_2"/>
not <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></i>;<br/>
<a NAME="E6:16_2_1_3_2"/><b>then </b>
not <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c4">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E7:16_2_1_3_2"/>
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r2</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E4:16_2_1_3_2"><i><font color="Green" title="E24">A39</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:16_2_1_3"/>
<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E2:16_2_1_3"><i><font color="Green" title="E23">A36</font></i></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E5:16_2_1"/>
<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E2:16_2_1"><i><font color="Green" title="E20">A25</font></i></a>, <a class="txt" href="hessenbe.html#E3:16_2_1"><i><font color="Green" title="E21">A29</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_2"/><i><font color="Green" title="E19">A40</font></i>: 
not <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><div><i><font color="Green" title="E20">A41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_2_1"/><i><font color="Green" title="E21">A42</font></i>: 
( <font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or <font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or <font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><div><i><font color="Green" title="E22">A43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_2_1_1"/><i><font color="Green" title="E23">A44</font></i>: 
<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><a NAME="E2:16_2_2_1_1"/>
( <font color="Maroon" title="c7">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r2</font> &amp; <font color="Maroon" title="c7">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r3</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_2_2_1_1_1"/>
<font color="Maroon" title="c7">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r2</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r3</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_2_1_1_1_1"/>
<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_2_1_1"><i><font color="Green" title="E23">A44</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E2:16_2_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E24">A45</font></i>: 
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E3:16_2_2_1_1_1_1"/>
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q3</font>
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16_2_2"><i><font color="Green" title="E19">A40</font></i></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></i>;<br/>
<a NAME="E4:16_2_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E2:16_2_2_1_1_1_1"><i><font color="Green" title="E24">A45</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E5:16_2_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E25">A46</font></i>: 
<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E6:16_2_2_1_1_1_1"/>
not <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></i>;<br/>
<a NAME="E7:16_2_2_1_1_1_1"/><b>then </b>
( not <font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E8:16_2_2_1_1_1_1"/>
<font color="Maroon" title="c7">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r2</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E5:16_2_2_1_1_1_1"><i><font color="Green" title="E25">A46</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:16_2_2_1_1_1"/>
<font color="Maroon" title="c7">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r3</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_2_1_1_1_2"/><i><font color="Green" title="E24">A47</font></i>: 
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q2</font>
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16_2"><i><font color="Green" title="E17">A22</font></i></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></i>;<br/>
<a NAME="E2:16_2_2_1_1_1_2"/>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16_2_2_1_1"><i><font color="Green" title="E23">A44</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E3:16_2_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c8">r3</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_2_1_1_1_2"><i><font color="Green" title="E24">A47</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E4:16_2_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E25">A48</font></i>: 
<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E5:16_2_2_1_1_1_2"/>
not <font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E6:16_2_2_1_1_1_2"/>
<font color="Maroon" title="c7">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r3</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E4:16_2_2_1_1_1_2"><i><font color="Green" title="E25">A48</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E3:16_2_2_1_1"/>
<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><i><font color="Green" title="E23">A49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_2_1_2"/><i><font color="Green" title="E24">A50</font></i>: 
<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><a NAME="E2:16_2_2_1_2"/>
( <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r3</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_2_2_1_2_1"/>
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r3</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r1</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_2_1_2_1_1"/><i><font color="Green" title="E25">A51</font></i>: 
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">q1</font>
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></i>;<br/>
<a NAME="E2:16_2_2_1_2_1_1"/>
( <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_2_1_2"><i><font color="Green" title="E24">A50</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E3:16_2_2_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c8">r3</font>,<font color="Maroon" title="c5">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16_2_2_1_2_1_1"><i><font color="Green" title="E25">A51</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E4:16_2_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E26">A52</font></i>: 
( <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E5:16_2_2_1_2_1_1"/>
not <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16_2"><i><font color="Green" title="E17">A22</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E6:16_2_2_1_2_1_1"/>
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r3</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E4:16_2_2_1_2_1_1"><i><font color="Green" title="E26">A52</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:16_2_2_1_2_1"/>
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r1</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_2_1_2_1_2"/><i><font color="Green" title="E25">A53</font></i>: 
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q3</font>
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16_2_2"><i><font color="Green" title="E19">A40</font></i></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></i>;<br/>
<a NAME="E2:16_2_2_1_2_1_2"/>
( <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_2_1_2"><i><font color="Green" title="E24">A50</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E3:16_2_2_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E4:16_2_2_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c5">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E5:16_2_2_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c5">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_2_1_2_1_2"><i><font color="Green" title="E25">A53</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E6:16_2_2_1_2_1_2"/><b>then </b><i><font color="Green" title="E26">A54</font></i>: 
( <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E7:16_2_2_1_2_1_2"/>
( not <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2"><i><font color="Green" title="E17">A22</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E8:16_2_2_1_2_1_2"/><b>then </b>
not <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></i>;<br/>
<a NAME="E9:16_2_2_1_2_1_2"/><b>then </b>
not <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E10:16_2_2_1_2_1_2"/>
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r1</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E6:16_2_2_1_2_1_2"><i><font color="Green" title="E26">A54</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E3:16_2_2_1_2"/>
<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_2_1_3"/><i><font color="Green" title="E24">A55</font></i>: 
<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><a NAME="E2:16_2_2_1_3"/>
( <font color="Maroon" title="c6">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r2</font> &amp; <font color="Maroon" title="c6">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_3_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_2_2_1_3_1"/>
<font color="Maroon" title="c6">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r2</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r1</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_2_1_3_1_1"/>
<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_2_1_3"><i><font color="Green" title="E24">A55</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E2:16_2_2_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E3:16_2_2_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E4:16_2_2_1_3_1_1"/><b>then </b><i><font color="Green" title="E25">A56</font></i>: 
( <font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E5:16_2_2_1_3_1_1"/>
not <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_2"><i><font color="Green" title="E19">A40</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></i>;<br/>
<a NAME="E6:16_2_2_1_3_1_1"/><b>then </b>
not <font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E7:16_2_2_1_3_1_1"/>
<font color="Maroon" title="c6">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r2</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E4:16_2_2_1_3_1_1"><i><font color="Green" title="E25">A56</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:16_2_2_1_3_1"/>
<font color="Maroon" title="c6">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r1</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_3_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_2_1_3_1_2"/>
( <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_2_1_3"><i><font color="Green" title="E24">A55</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E2:16_2_2_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E3:16_2_2_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E4:16_2_2_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2"><i><font color="Green" title="E17">A22</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E5:16_2_2_1_3_1_2"/><b>then </b><i><font color="Green" title="E25">A57</font></i>: 
( <font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c10">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E6:16_2_2_1_3_1_2"/>
( not <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_2"><i><font color="Green" title="E19">A40</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E7:16_2_2_1_3_1_2"/><b>then </b>
not <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></i>;<br/>
<a NAME="E8:16_2_2_1_3_1_2"/><b>then </b>
not <font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E9:16_2_2_1_3_1_2"/>
<font color="Maroon" title="c6">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r1</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E5:16_2_2_1_3_1_2"><i><font color="Green" title="E25">A57</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E3:16_2_2_1_3"/>
<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E5:16_2_2_1"/>
<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16_2_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="hessenbe.html#E2:16_2_2_1"><i><font color="Green" title="E22">A43</font></i></a>, <a class="txt" href="hessenbe.html#E3:16_2_2_1"><i><font color="Green" title="E23">A49</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_2_2"/><i><font color="Green" title="E21">A58</font></i>: 
( not <font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><b>consider </b><font color="Maroon" title="c11">o</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PCPP</font><b> such that </b><br/><a NAME="E3:16_2_2_2"/><i><font color="Green" title="E22">A59</font></i>: 
( <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c11">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c11">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D14" title="ANPROJ_2:def.14">ANPROJ_2:def 14</a></i>;<br/><a NAME="E4:16_2_2_2"/><i><font color="Green" title="E23">A60</font></i>: 
( <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_2_2_1"/>
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c11">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E2:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="hessenbe.html#E3:16_2_2_2"><i><font color="Green" title="E22">A59</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E2:16_2_2_2_1"/>
<font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c3">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/>

<a NAME="E3:16_2_2_2_1"/>
<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c6">q3</font>,<font color="Maroon" title="c11">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E3:16_2_2_2"><i><font color="Green" title="E22">A59</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E4:16_2_2_2_1"/>
<font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c6">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/>

<b>thus </b><a NAME="E5:16_2_2_2_1"/>
( <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="hessenbe.html#E3:16_2_2_2"><i><font color="Green" title="E22">A59</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E5:16_2_2_2"/>
not <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_2_2_2"/>
not <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_2_2"><i><font color="Green" title="E21">A58</font></i></a>, <a class="txt" href="hessenbe.html#E3:16_2_2_2"><i><font color="Green" title="E22">A59</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E2:16_2_2_2_2"/>
not <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:16_2_2_2"/>
<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hessenbe.html#E2:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2"><i><font color="Green" title="E17">A22</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_2"><i><font color="Green" title="E19">A40</font></i></a>, <a class="txt" href="hessenbe.html#E1:16_2_2_2"><i><font color="Green" title="E21">A58</font></i></a>, <a class="txt" href="hessenbe.html#E3:16_2_2_2"><i><font color="Green" title="E22">A59</font></i></a>, <a class="txt" href="hessenbe.html#E4:16_2_2_2"><i><font color="Green" title="E23">A60</font></i></a>, <a class="ref" href="anproj_2.html#D13" title="ANPROJ_2:def.13">ANPROJ_2:def 13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E4:16_2_2"/>
<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E2:16_2_2"><i><font color="Green" title="E20">A41</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E4:16_2"/>
<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E2:16_2"><i><font color="Green" title="E18">A23</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:16"/>
<font color="Maroon" title="c10">r1</font>,<font color="Maroon" title="c9">r2</font>,<font color="Maroon" title="c8">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E3:16"><i><font color="Green" title="E16">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
