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<b>let </b><font color="Maroon" title="c1">V</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</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">p1</font>, <font color="Olive" title="b2">r2</font>, <font color="Olive" title="b3">q</font>, <font color="Olive" title="b4">r1</font>, <font color="Olive" title="b5">q1</font>, <font color="Olive" title="b6">p</font>, <font color="Olive" title="b7">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>  st <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">r2</font>,<font color="Olive" title="b3">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b4">r1</font>,<font color="Olive" title="b5">q1</font>,<font color="Olive" title="b3">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b4">r1</font>,<font color="Olive" title="b6">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">r2</font>,<font color="Olive" title="b5">q1</font>,<font color="Olive" title="b6">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b5">q1</font>,<font color="Olive" title="b7">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">r2</font>,<font color="Olive" title="b4">r1</font>,<font color="Olive" title="b7">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b6">p</font>,<font color="Olive" title="b3">q</font>,<font color="Olive" title="b7">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">r2</font>,<font color="Olive" title="b5">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">r2</font>,<font color="Olive" title="b4">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b4">r1</font>,<font color="Olive" title="b5">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/><font color="Olive" title="b2">r2</font>,<font color="Olive" title="b4">r1</font>,<font color="Olive" title="b5">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><b>let </b><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">r2</font>, <font color="Maroon" title="c4">q</font>, <font color="Maroon" title="c5">r1</font>, <font color="Maroon" title="c6">q1</font>, <font color="Maroon" title="c7">p</font>, <font color="Maroon" title="c8">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">r2</font>,<font color="Maroon" title="c4">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c4">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c7">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">r2</font>,<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c8">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">r2</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c8">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c8">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">r2</font>,<font color="Maroon" title="c6">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">r2</font>,<font color="Maroon" title="c5">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c6">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  implies <font color="Maroon" title="c3">r2</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c6">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/>















<b>assume </b><a NAME="E1:84"/><i><font color="Green" title="E74">A1</font></i>: 
( <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">r2</font>,<font color="Maroon" title="c4">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c4">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c7">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">r2</font>,<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c8">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">r2</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c8">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c8">r</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="c2">p1</font>,<font color="Maroon" title="c3">r2</font>,<font color="Maroon" title="c6">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">r2</font>,<font color="Maroon" title="c5">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c6">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or <font color="Maroon" title="c3">r2</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c6">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/>

<b>then consider </b><font color="Maroon" title="c9">u</font>, <font color="Maroon" title="c10">v</font>, <font color="Maroon" title="c11">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:84"/><i><font color="Green" title="E75">A2</font></i>: 
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c9">u</font> &amp; <font color="Maroon" title="c3">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c10">v</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c11">z</font> &amp; not <font color="Maroon" title="c9">u</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c10">v</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c11">z</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c11">z</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">w</font>, <font color="Maroon" title="c13">y</font>, <font color="Maroon" title="c14">z1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E5:84"/><i><font color="Green" title="E76">A3</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c12">w</font> &amp; <font color="Maroon" title="c6">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c13">y</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c14">z1</font> &amp; not <font color="Maroon" title="c12">w</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c13">y</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c14">z1</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c12">w</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c14">z1</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="#E1:84"><i><font color="Green" title="E74">A1</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">u1</font>, <font color="Maroon" title="c16">w1</font>, <font color="Maroon" title="c17">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E7:84"/><i><font color="Green" title="E77">A4</font></i>: 
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c15">u1</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c16">w1</font> &amp; <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c17">x</font> &amp; not <font color="Maroon" title="c15">u1</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c16">w1</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c17">x</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c15">u1</font>,<font color="Maroon" title="c16">w1</font>,<font color="Maroon" title="c17">x</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="#E1:84"><i><font color="Green" title="E74">A1</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">v1</font>, <font color="Maroon" title="c19">y1</font>, <font color="Maroon" title="c20">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E9:84"/><i><font color="Green" title="E78">A5</font></i>: 
( <font color="Maroon" title="c3">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c18">v1</font> &amp; <font color="Maroon" title="c6">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c19">y1</font> &amp; <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c20">x1</font> &amp; not <font color="Maroon" title="c18">v1</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c19">y1</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c20">x1</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c18">v1</font>,<font color="Maroon" title="c19">y1</font>,<font color="Maroon" title="c20">x1</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="#E1:84"><i><font color="Green" title="E74">A1</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">u2</font>, <font color="Maroon" title="c22">y2</font>, <font color="Maroon" title="c23">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E11:84"/><i><font color="Green" title="E79">A6</font></i>: 
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c21">u2</font> &amp; <font color="Maroon" title="c6">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c22">y2</font> &amp; <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c23">t</font> &amp; not <font color="Maroon" title="c21">u2</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c22">y2</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c23">t</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c21">u2</font>,<font color="Maroon" title="c22">y2</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="#E1:84"><i><font color="Green" title="E74">A1</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">v2</font>, <font color="Maroon" title="c25">w2</font>, <font color="Maroon" title="c26">t1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E13:84"/><i><font color="Green" title="E80">A7</font></i>: 
( <font color="Maroon" title="c3">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c24">v2</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c25">w2</font> &amp; <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c26">t1</font> &amp; not <font color="Maroon" title="c24">v2</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c25">w2</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c26">t1</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c24">v2</font>,<font color="Maroon" title="c25">w2</font>,<font color="Maroon" title="c26">t1</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="#E1:84"><i><font color="Green" title="E74">A1</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c27">x2</font>, <font color="Maroon" title="c28">z2</font>, <font color="Maroon" title="c29">t2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E15:84"/><i><font color="Green" title="E81">A8</font></i>: 
( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c27">x2</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c28">z2</font> &amp; <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c29">t2</font> &amp; not <font color="Maroon" title="c27">x2</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c28">z2</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c29">t2</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c27">x2</font>,<font color="Maroon" title="c28">z2</font>,<font color="Maroon" title="c29">t2</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="#E1:84"><i><font color="Green" title="E74">A1</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<a NAME="E16:84"/>
( <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c11">z</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c12">w</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c11">z</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c12">w</font>,<font color="Maroon" title="c17">x</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c17">x</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c12">w</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<a NAME="E1:84_1"/><i><font color="Green" title="E82">A9</font></i>: 
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c14">z1</font>,<font color="Maroon" title="c11">z</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c16">w1</font>,<font color="Maroon" title="c12">w</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c15">u1</font>,<font color="Maroon" title="c9">u</font> )
 
<b>by </b><i><a class="txt" href="#E3:84"><i><font color="Green" title="E75">A2</font></i></a>, <a class="txt" href="#E5:84"><i><font color="Green" title="E76">A3</font></i></a>, <a class="txt" href="#E7:84"><i><font color="Green" title="E77">A4</font></i></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></i>;<br/>
<a NAME="E2:84_1"/><i><font color="Green" title="E83">A10</font></i>: 
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c19">y1</font>,<font color="Maroon" title="c13">y</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c20">x1</font>,<font color="Maroon" title="c17">x</font> )
 
<b>by </b><i><a class="txt" href="#E5:84"><i><font color="Green" title="E76">A3</font></i></a>, <a class="txt" href="#E7:84"><i><font color="Green" title="E77">A4</font></i></a>, <a class="txt" href="#E9:84"><i><font color="Green" title="E78">A5</font></i></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></i>;<br/>
<a NAME="E3:84_1"/><i><font color="Green" title="E84">A11</font></i>: 
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c21">u2</font>,<font color="Maroon" title="c9">u</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c22">y2</font>,<font color="Maroon" title="c13">y</font> )
 
<b>by </b><i><a class="txt" href="#E3:84"><i><font color="Green" title="E75">A2</font></i></a>, <a class="txt" href="#E5:84"><i><font color="Green" title="E76">A3</font></i></a>, <a class="txt" href="#E11:84"><i><font color="Green" title="E79">A6</font></i></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></i>;<br/>
<a NAME="E4:84_1"/><i><font color="Green" title="E85">A12</font></i>: 
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c24">v2</font>,<font color="Maroon" title="c10">v</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c25">w2</font>,<font color="Maroon" title="c12">w</font> )
 
<b>by </b><i><a class="txt" href="#E3:84"><i><font color="Green" title="E75">A2</font></i></a>, <a class="txt" href="#E5:84"><i><font color="Green" title="E76">A3</font></i></a>, <a class="txt" href="#E13:84"><i><font color="Green" title="E80">A7</font></i></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></i>;<br/>
<a NAME="E5:84_1"/><i><font color="Green" title="E86">A13</font></i>: 
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c26">t1</font>,<font color="Maroon" title="c23">t</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c27">x2</font>,<font color="Maroon" title="c17">x</font> )
 
<b>by </b><i><a class="txt" href="#E7:84"><i><font color="Green" title="E77">A4</font></i></a>, <a class="txt" href="#E11:84"><i><font color="Green" title="E79">A6</font></i></a>, <a class="txt" href="#E13:84"><i><font color="Green" title="E80">A7</font></i></a>, <a class="txt" href="#E15:84"><i><font color="Green" title="E81">A8</font></i></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></i>;<br/>
<a NAME="E6:84_1"/><i><font color="Green" title="E87">A14</font></i>: 
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c28">z2</font>,<font color="Maroon" title="c11">z</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c29">t2</font>,<font color="Maroon" title="c23">t</font> )
 
<b>by </b><i><a class="txt" href="#E3:84"><i><font color="Green" title="E75">A2</font></i></a>, <a class="txt" href="#E11:84"><i><font color="Green" title="E79">A6</font></i></a>, <a class="txt" href="#E15:84"><i><font color="Green" title="E81">A8</font></i></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></i>;<br/>
<a NAME="E7:84_1"/><i><font color="Green" title="E88">A15</font></i>: 
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c18">v1</font>,<font color="Maroon" title="c10">v</font>
 
<b>by </b><i><a class="txt" href="#E3:84"><i><font color="Green" title="E75">A2</font></i></a>, <a class="txt" href="#E9:84"><i><font color="Green" title="E78">A5</font></i></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></i>;<br/>
<b>thus </b><a NAME="E8:84_1"/>
<font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c11">z</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="#E3:84"><i><font color="Green" title="E75">A2</font></i></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="c12">w</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c11">z</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c12">w</font>,<font color="Maroon" title="c17">x</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c17">x</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c12">w</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )</span><br/>

<b>thus </b><a NAME="E9:84_1"/>
<font color="Maroon" title="c12">w</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c11">z</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="#E5:84"><i><font color="Green" title="E76">A3</font></i></a>, <a class="txt" href="#E1:84_1"><i><font color="Green" title="E82">A9</font></i></a>, <a class="ref" href="anproj_1.html#T9" title="ANPROJ_1:th.9">ANPROJ_1:9</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="c9">u</font>,<font color="Maroon" title="c12">w</font>,<font color="Maroon" title="c17">x</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c17">x</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c12">w</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )</span><br/>

<b>thus </b><a NAME="E10:84_1"/>
<font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c12">w</font>,<font color="Maroon" title="c17">x</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="#E7:84"><i><font color="Green" title="E77">A4</font></i></a>, <a class="txt" href="#E1:84_1"><i><font color="Green" title="E82">A9</font></i></a>, <a class="ref" href="anproj_1.html#T9" title="ANPROJ_1:th.9">ANPROJ_1:9</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="c10">v</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c17">x</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c12">w</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )</span><br/>

<b>thus </b><a NAME="E11:84_1"/>
<font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c17">x</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="#E9:84"><i><font color="Green" title="E78">A5</font></i></a>, <a class="txt" href="#E2:84_1"><i><font color="Green" title="E83">A10</font></i></a>, <a class="txt" href="#E7:84_1"><i><font color="Green" title="E88">A15</font></i></a>, <a class="ref" href="anproj_1.html#T9" title="ANPROJ_1:th.9">ANPROJ_1:9</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="c9">u</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c12">w</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )</span><br/>

<b>thus </b><a NAME="E12:84_1"/>
<font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="#E11:84"><i><font color="Green" title="E79">A6</font></i></a>, <a class="txt" href="#E3:84_1"><i><font color="Green" title="E84">A11</font></i></a>, <a class="ref" href="anproj_1.html#T9" title="ANPROJ_1:th.9">ANPROJ_1:9</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="c10">v</font>,<font color="Maroon" title="c12">w</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )</span><br/>

<b>thus </b><a NAME="E13:84_1"/>
<font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c12">w</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="#E13:84"><i><font color="Green" title="E80">A7</font></i></a>, <a class="txt" href="#E4:84_1"><i><font color="Green" title="E85">A12</font></i></a>, <a class="txt" href="#E5:84_1"><i><font color="Green" title="E86">A13</font></i></a>, <a class="ref" href="anproj_1.html#T9" title="ANPROJ_1:th.9">ANPROJ_1:9</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="c17">x</font>,<font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> </span><br/>

<b>thus </b><a NAME="E14:84_1"/>
<font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c23">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="#E15:84"><i><font color="Green" title="E81">A8</font></i></a>, <a class="txt" href="#E5:84_1"><i><font color="Green" title="E86">A13</font></i></a>, <a class="txt" href="#E6:84_1"><i><font color="Green" title="E87">A14</font></i></a>, <a class="ref" href="anproj_1.html#T9" title="ANPROJ_1:th.9">ANPROJ_1:9</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="E17:84"/><b>then </b>
( <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c13">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  or <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c12">w</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  or <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c12">w</font>,<font color="Maroon" title="c13">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  or <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c12">w</font>,<font color="Maroon" title="c13">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 
<b>by </b><i><a class="txt" href="#E3:84"><i><font color="Green" title="E75">A2</font></i></a>, <a class="txt" href="#E7:84"><i><font color="Green" title="E77">A4</font></i></a>, <a class="txt" href="#E11:84"><i><font color="Green" title="E79">A6</font></i></a>, <a class="ref" href="anproj_1.html#T23" title="ANPROJ_1:th.23">ANPROJ_1:23</a></i>;<br/>
<b>hence </b><a NAME="E18:84"/>
( <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">r2</font>,<font color="Maroon" title="c6">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">r2</font>,<font color="Maroon" title="c5">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c6">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or <font color="Maroon" title="c3">r2</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c6">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E3:84"><i><font color="Green" title="E75">A2</font></i></a>, <a class="txt" href="#E5:84"><i><font color="Green" title="E76">A3</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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