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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">o</font>, <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">a1</font>, <font color="Olive" title="b4">b2</font>, <font color="Olive" title="b5">a2</font>, <font color="Olive" title="b6">b3</font>, <font color="Olive" title="b7">a3</font>, <font color="Olive" title="b8">c1</font>, <font color="Olive" title="b9">c3</font>, <font color="Olive" title="b10">c2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PCPP</font>  st <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Olive" title="b3">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">b2</font> &amp; <font color="Olive" title="b5">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">b2</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">b3</font> &amp; <font color="Olive" title="b7">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">b3</font> &amp; not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">a1</font>,<font color="Olive" title="b5">a2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">a1</font>,<font color="Olive" title="b7">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b5">a2</font>,<font color="Olive" title="b7">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b8">c1</font>,<font color="Olive" title="b9">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">a1</font>,<font color="Olive" title="b5">a2</font>,<font color="Olive" title="b9">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">b1</font>,<font color="Olive" title="b4">b2</font>,<font color="Olive" title="b9">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b5">a2</font>,<font color="Olive" title="b7">a3</font>,<font color="Olive" title="b8">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b4">b2</font>,<font color="Olive" title="b6">b3</font>,<font color="Olive" title="b8">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">a1</font>,<font color="Olive" title="b7">a3</font>,<font color="Olive" title="b10">c2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">b1</font>,<font color="Olive" title="b6">b3</font>,<font color="Olive" title="b10">c2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">a1</font>,<font color="Olive" title="b2">b1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b5">a2</font>,<font color="Olive" title="b4">b2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b7">a3</font>,<font color="Olive" title="b6">b3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/><font color="Olive" title="b8">c1</font>,<font color="Olive" title="b10">c2</font>,<font color="Olive" title="b9">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><b>let </b><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c4">a1</font>, <font color="Maroon" title="c5">b2</font>, <font color="Maroon" title="c6">a2</font>, <font color="Maroon" title="c7">b3</font>, <font color="Maroon" title="c8">a3</font>, <font color="Maroon" title="c9">c1</font>, <font color="Maroon" title="c10">c3</font>, <font color="Maroon" title="c11">c2</font> 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">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b1</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b1</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b3</font> &amp; <font color="Maroon" title="c8">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b3</font> &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c6">a2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c9">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c9">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c11">c2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c11">c2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c3">b1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c5">b2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c7">b3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  implies <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/>





















<b>assume </b><b>that </b><br/><a NAME="E1:19"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b1</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b1</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b3</font> &amp; <font color="Maroon" title="c8">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b3</font> )
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green" title="E18">A2</font></i>: 
( not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c6">a2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>and </b><br/><a NAME="E3:19"/><i><font color="Green" title="E19">A3</font></i>: 
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>and </b><br/><a NAME="E4:19"/><i><font color="Green" title="E20">A4</font></i>: 
( <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c9">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c9">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c11">c2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c11">c2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>and </b><br/><a NAME="E5:19"/><i><font color="Green" title="E21">A5</font></i>: 
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c3">b1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c5">b2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c7">b3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/>

<a NAME="E6:19"/><i><font color="Green" title="E22">A6</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a1</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a2</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a2</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a3</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a3</font> &amp; <font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a3</font> )
 
<b>by </b><i><a class="txt" href="#E2:19"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1"/><i><font color="Green" title="E23">A7</font></i>: 
( not <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c7">b3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  implies <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><b>consider </b><font color="Maroon" title="c12">x</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="E3:19_1"/><i><font color="Green" title="E24">A8</font></i>: 
( <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c12">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c12">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D14" title="ANPROJ_2:def.14">ANPROJ_2:def 14</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">y</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="E5:19_1"/><i><font color="Green" title="E25">A9</font></i>: 
( <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c13">y</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c13">y</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/><b>consider </b><font color="Maroon" title="c14">a</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="E7:19_1"/><i><font color="Green" title="E26">A10</font></i>: 
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c14">a</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c14">a</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/><b>consider </b><font color="Maroon" title="c15">z</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="E9:19_1"/><i><font color="Green" title="E27">A11</font></i>: 
( <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c15">z</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c15">z</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/><b>assume </b><a NAME="E10:19_1"/><i><font color="Green" title="E28">A12</font></i>: 
not <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><div><i><font color="Green" title="E29">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1_1"/><i><font color="Green" title="E30">A14</font></i>: 
( <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c12">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c13">y</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"> contradiction</span><br/><b>thus </b><a NAME="E2:19_1_1"/>
contradiction
  <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="19_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_1_1"/><i><font color="Green" title="E31">A15</font></i>: 
( <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font> &amp; <font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font> &amp; <font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c1</font> &amp; <font color="Maroon" title="c8">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c1</font> &amp; <font color="Maroon" title="c9">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_1_1_1"/><i><font color="Green" title="E32">A16</font></i>: 
<font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_1_1_1_1"/>
( not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c4">a1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="#E2:19"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E4:19"><i><font color="Green" title="E20">A4</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="E2:19_1_1_1_1_1"/>
<font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font>
 <b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E5:19"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="hessenbe.html#T10" target="_self" title="HESSENBE:th.10">Th10</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="E2:19_1_1_1_1"/>
( not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c6">a2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c9">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c9">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="#E2:19"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E4:19"><i><font color="Green" title="E20">A4</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="E3:19_1_1_1_1"/>
( <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font> &amp; <font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font> &amp; <font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c1</font> &amp; <font color="Maroon" title="c8">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c1</font> &amp; <font color="Maroon" title="c9">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font> )
 <b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E2:19"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E3:19"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E4:19"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="#E5:19"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="#E1:19_1_1_1_1"><i><font color="Green" title="E32">A16</font></i></a>, <a class="ref" href="hessenbe.html#T10" target="_self" title="HESSENBE:th.10">Th10</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>
<a NAME="E2:19_1_1_1"/><i><font color="Green" title="E32">A17</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">a</font> &amp; <font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">a</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a2</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a3</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c8">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_1_1_2"/><i><font color="Green" title="E33">A18</font></i>: 
<font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font>
 
<b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E2:19"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E4:19"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="#E5:19"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="hessenbe.html#T10" target="_self" title="HESSENBE:th.10">Th10</a></i>;<br/>
<a NAME="E2:19_1_1_1_2"/><i><font color="Green" title="E34">A19</font></i>: 
<font color="Maroon" title="c9">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font>
 
<b>by </b><i><a class="txt" href="#E3:19"><i><font color="Green" title="E19">A3</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="E3:19_1_1_1_2"/>
( not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c6">a2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c9">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c9">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="#E2:19"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E4:19"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E4:19_1_1_1_2"/><b>then </b><i><font color="Green" title="E35">A20</font></i>: 
<font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c1</font>
 
<b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E5:19"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="hessenbe.html#T10" target="_self" title="HESSENBE:th.10">Th10</a></i>;<br/>
<a NAME="E5:19_1_1_1_2"/><i><font color="Green" title="E36">A21</font></i>: 
<font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1_1_1_2_1"/><i><font color="Green" title="E37">A22</font></i>: 
not <font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:19_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c9">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c4">a1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="#E4:19"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="#E7:19_1"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E3:19_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c4">a1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="#E1:19_1_1_1_2"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="#E4:19_1_1_1_2"><i><font color="Green" title="E35">A20</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/>
<a NAME="E4:19_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c4">a1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<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="E5:19_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:19_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="#E7:19_1"><i><font color="Green" title="E26">A10</font></i></a>, <a class="txt" href="#E2:19_1_1_1_2"><i><font color="Green" title="E34">A19</font></i></a>, <a class="txt" href="#E1:19_1_1_1_2_1"><i><font color="Green" title="E37">A22</font></i></a>, <a class="ref" href="anproj_2.html#D8" title="ANPROJ_2:def.8">ANPROJ_2:def 8</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="E6:19_1_1_1_2"/><i><font color="Green" title="E37">A23</font></i>: 
<font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_1_1_2_2"/>
( not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c4">a1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="#E2:19"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E4:19"><i><font color="Green" title="E20">A4</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="E2:19_1_1_1_2_2"/>
<font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c3</font>
 <b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E5:19"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="hessenbe.html#T10" target="_self" title="HESSENBE:th.10">Th10</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="E7:19_1_1_1_2"/>
<font color="Maroon" title="c8">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c1</font>
 
<b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E2:19"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E4:19"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="#E5:19"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="hessenbe.html#T10" target="_self" title="HESSENBE:th.10">Th10</a></i>;<br/>
<b>hence </b><a NAME="E8:19_1_1_1_2"/>
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">a</font> &amp; <font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">a</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a2</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a3</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c8">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">z</font> )
 <b>by </b><i><a class="txt" href="#E2:19"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E3:19"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E4:19"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="#E1:19_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="#E7:19_1"><i><font color="Green" title="E26">A10</font></i></a>, <a class="txt" href="#E9:19_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="txt" href="#E1:19_1_1_1_2"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="#E4:19_1_1_1_2"><i><font color="Green" title="E35">A20</font></i></a>, <a class="txt" href="#E5:19_1_1_1_2"><i><font color="Green" title="E36">A21</font></i></a>, <a class="txt" href="#E6:19_1_1_1_2"><i><font color="Green" title="E37">A23</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a>, <a class="ref" href="hessenbe.html#T12" target="_self" title="HESSENBE:th.12">Th12</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>
<a NAME="E3:19_1_1_1"/><i><font color="Green" title="E33">A24</font></i>: 
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c9">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_1_1_3"/>
( not <font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c2">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c9">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c1</font> )
 
<b>by </b><i><a class="txt" href="#E2:19"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E4:19"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="#E1:19_1_1_1"><i><font color="Green" title="E31">A15</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E2:19_1_1_1_3"/><b>then </b>
not <font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c2">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1_1_1_3"/>
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c9">c1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <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>
<a NAME="E4:19_1_1_1"/><i><font color="Green" title="E34">A25</font></i>: 
( <font color="Maroon" title="c3">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Maroon" title="c3">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b3</font> &amp; <font color="Maroon" title="c5">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b3</font> )
 
<b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E2:19"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E5:19"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self" title="HESSENBE:th.9">Th9</a></i>;<br/>
<a NAME="E5:19_1_1_1"/><i><font color="Green" title="E35">A26</font></i>: 
<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c12">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="#E3:19_1"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="#E5:19_1"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="#E1:19_1_1"><i><font color="Green" title="E30">A14</font></i></a>, <a class="txt" href="#E1:19_1_1_1"><i><font color="Green" title="E31">A15</font></i></a>, <a class="txt" href="#E2:19_1_1_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="#E3:19_1_1_1"><i><font color="Green" title="E33">A24</font></i></a>, <a class="ref" href="hessenbe.html#T17" target="_self" title="HESSENBE:th.17">Th17</a></i>;<br/>
<a NAME="E6:19_1_1_1"/>
( not <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">b1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b1</font> )
 
<b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E2:19"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E5:19"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E7:19_1_1_1"/><b>then </b>
not <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T8" target="_self" title="HESSENBE:th.8">Th8</a></i>;<br/>
<a NAME="E8:19_1_1_1"/><b>then </b>
not <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c3">b1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/>
<a NAME="E9:19_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c11">c2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="#E4:19"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="#E6:19"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="#E3:19_1"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="#E1:19_1_1"><i><font color="Green" title="E30">A14</font></i></a>, <a class="txt" href="#E4:19_1_1_1"><i><font color="Green" title="E34">A25</font></i></a>, <a class="txt" href="#E5:19_1_1_1"><i><font color="Green" title="E35">A26</font></i></a>, <a class="ref" href="hessenbe.html#T17" target="_self" title="HESSENBE:th.17">Th17</a></i>;<br/>
<b>hence </b><a NAME="E10:19_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E10:19_1"><i><font color="Green" title="E28">A12</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><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1_2"/><i><font color="Green" title="E30">A27</font></i>: 
(  not <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c12">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or  not <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c13">y</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="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1_2_1"/><i><font color="Green" title="E31">A28</font></i>: 
not <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">c3</font>,<font color="Maroon" title="c13">y</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"> contradiction</span><br/><a NAME="E2:19_1_2_1"/><i><font color="Green" title="E32">A29</font></i>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c13">y</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="#E5:19"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="#E6:19"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="#E5:19_1"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self" title="HESSENBE:th.4">Th4</a></i>;<br/><a NAME="E3:19_1_2_1"/><i><font color="Green" title="E33">A30</font></i>: 
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">a1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c6">a2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">b3</font>,<font color="Maroon" title="c8">a3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E5:19"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self" title="HESSENBE:th.3">Th3</a></i>;<br/><a NAME="E4:19_1_2_1"/>
( not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">b2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c7">b3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c7">b3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E2:19"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E5:19"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="hessenbe.html#T19" target="_self" title="HESSENBE:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E5:19_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E3:19"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E4:19"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="#E6:19"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="#E5:19_1"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="#E10:19_1"><i><font color="Green" title="E28">A12</font></i></a>, <a class="txt" href="#E1:19_1_2_1"><i><font color="Green" title="E31">A28</font></i></a>, <a class="txt" href="#E2:19_1_2_1"><i><font color="Green" title="E32">A29</font></i></a>, <a class="txt" href="#E3:19_1_2_1"><i><font color="Green" title="E33">A30</font></i></a>, <a class="txt" href="hessenbe.html#E22"><i><font color="Green" title="E15">Lm2</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="E3:19_1_2"/>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E2:19"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E3:19"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E4:19"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="#E5:19"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="#E3:19_1"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="#E1:19_1_2"><i><font color="Green" title="E30">A27</font></i></a>, <a class="txt" href="hessenbe.html#E22"><i><font color="Green" title="E15">Lm2</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="E13:19_1"/>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="#E11:19_1"><i><font color="Green" title="E29">A13</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="E8:19"/>
<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">c3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E2:19"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E3:19"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E4:19"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="#E5:19"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="hessenbe.html#E21"><i><font color="Green" title="E14">Lm1</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>
