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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">p</font>, <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">r</font>, <font color="Olive" title="b5">r1</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 not <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b4">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font>,<font color="Olive" title="b5">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/> ex <font color="Olive" title="b6">r2</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 <br/>( <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">p2</font>,<font color="Olive" title="b6">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b4">r</font>,<font color="Olive" title="b5">r1</font>,<font color="Olive" title="b6">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font>, <font color="Maroon" title="c5">r</font>, <font color="Maroon" title="c6">r1</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"> ( not <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  implies  ex <font color="Olive" title="b1">r2</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 <br/>( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">r1</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  ) )</span><br/>











<b>assume </b><b>that </b><br/><a NAME="E1:70"/><i><font color="Green" title="E66">A1</font></i>: 
not <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>and </b><br/><a NAME="E2:70"/><i><font color="Green" title="E67">A2</font></i>: 
( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">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">  ex <font color="Olive" title="b1">r2</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 <br/>( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">r1</font>,<font color="Olive" title="b1">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/>

<b>consider </b><font color="Maroon" title="c7">u</font>, <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c9">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="E4:70"/><i><font color="Green" title="E68">A3</font></i>: 
( <font color="Maroon" title="c2">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="c7">u</font> &amp; <font color="Maroon" title="c3">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="c8">v</font> &amp; <font color="Maroon" title="c5">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="c9">t</font> &amp; not <font color="Maroon" title="c7">u</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c8">v</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c9">t</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">t</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="#E2:70"><i><font color="Green" title="E67">A2</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="c10">v1</font>, <font color="Maroon" title="c11">w</font>, <font color="Maroon" title="c12">s</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="E6:70"/><i><font color="Green" title="E69">A4</font></i>: 
( <font color="Maroon" title="c3">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="c10">v1</font> &amp; <font color="Maroon" title="c4">p2</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">w</font> &amp; <font color="Maroon" title="c6">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">s</font> &amp; not <font color="Maroon" title="c10">v1</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c11">w</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c12">s</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">s</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="#E2:70"><i><font color="Green" title="E67">A2</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<a NAME="E7:70"/>
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c8">v</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c11">w</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c12">s</font> )
 
<b>by </b><i><a class="txt" href="#E4:70"><i><font color="Green" title="E68">A3</font></i></a>, <a class="txt" href="#E6:70"><i><font color="Green" title="E69">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="E8:70"/><b>then </b><i><font color="Green" title="E70">A5</font></i>: 
<font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">s</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<b>by </b><i><a class="txt" href="#E6:70"><i><font color="Green" title="E69">A4</font></i></a>, <a class="ref" href="anproj_1.html#T9" title="ANPROJ_1:th.9">ANPROJ_1:9</a></i>;<br/>
<a NAME="E9:70"/>
not <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c11">w</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<b>by </b><i><a class="txt" href="#E1:70"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="#E4:70"><i><font color="Green" title="E68">A3</font></i></a>, <a class="txt" href="#E6:70"><i><font color="Green" title="E69">A4</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<b>then 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">V</font><b> such that </b><br/><a NAME="E11:70"/><i><font color="Green" title="E71">A6</font></i>: 
( <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c13">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c9">t</font>,<font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c13">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; not <font color="Maroon" title="c13">y</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> )
 <b>by </b><i><a class="txt" href="#E4:70"><i><font color="Green" title="E68">A3</font></i></a>, <a class="txt" href="#E8:70"><i><font color="Green" title="E70">A5</font></i></a>, <a class="ref" href="anproj_1.html#T20" title="ANPROJ_1:th.20">ANPROJ_1:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">r2</font> =  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c13">y</font> as   <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> <b>by </b><i><a class="txt" href="#E11:70"><i><font color="Green" title="E71">A6</font></i></a>, <a class="ref" href="anproj_1.html#T42" title="ANPROJ_1:th.42">ANPROJ_1:42</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">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="c2">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c14">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">r1</font>,<font color="Maroon" title="c14">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/>

<b>thus </b><a NAME="E13:70"/>
( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c14">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">r1</font>,<font color="Maroon" title="c14">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E4:70"><i><font color="Green" title="E68">A3</font></i></a>, <a class="txt" href="#E6:70"><i><font color="Green" title="E69">A4</font></i></a>, <a class="txt" href="#E11:70"><i><font color="Green" title="E71">A6</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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