<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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#K5" title="ANPROJ_1:func.5">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/><span class="kw">let </span><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#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:71"/><span class="lab"><font color="Green" title="E66">A1</font></span>: 
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> 
 <span class="kw">and </span><br/><a NAME="E2:71"/><span class="lab"><font color="Green" title="E67">A2</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E3:71"/><span class="lab"><font color="Green" title="E68">A3</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:71"/><span class="lab"><font color="Green" title="E69">A4</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Maroon" title="c7">u</font>
 <span class="kw">and </span><br/><a NAME="E6:71"/><span class="lab"><font color="Green" title="E70">A5</font></span>: 
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Maroon" title="c8">v</font>
 <span class="kw">and </span><br/><a NAME="E7:71"/><span class="lab"><font color="Green" title="E71">A6</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Maroon" title="c9">t</font>
 <span class="kw">and </span><br/><a NAME="E8:71"/><span class="lab"><font color="Green" title="E72">A7</font></span>: 
not <font color="Maroon" title="c7">u</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E9:71"/><span class="lab"><font color="Green" title="E73">A8</font></span>: 
not <font color="Maroon" title="c8">v</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E10:71"/><span class="lab"><font color="Green" title="E74">A9</font></span>: 
not <font color="Maroon" title="c9">t</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E11:71"/><span class="lab"><font color="Green" title="E75">A10</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E13:71"/><span class="lab"><font color="Green" title="E76">A11</font></span>: 
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Maroon" title="c10">v1</font>
 <span class="kw">and </span><br/><a NAME="E14:71"/><span class="lab"><font color="Green" title="E77">A12</font></span>: 
<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Maroon" title="c11">w</font>
 <span class="kw">and </span><br/><a NAME="E15:71"/><span class="lab"><font color="Green" title="E78">A13</font></span>: 
<font color="Maroon" title="c6">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Maroon" title="c12">s</font>
 <span class="kw">and </span><br/><a NAME="E16:71"/><span class="lab"><font color="Green" title="E79">A14</font></span>: 
not <font color="Maroon" title="c10">v1</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E17:71"/><span class="lab"><font color="Green" title="E80">A15</font></span>: 
not <font color="Maroon" title="c11">w</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E18:71"/><span class="lab"><font color="Green" title="E81">A16</font></span>: 
not <font color="Maroon" title="c12">s</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E19:71"/><span class="lab"><font color="Green" title="E82">A17</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71"><span class="lab"><font color="Green" title="E68">A3</font></span></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></span>;<br/>
<a NAME="E20:71"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:71"><span class="lab"><font color="Green" title="E70">A5</font></span></a>, <a class="txt" href="#E9:71"><span class="lab"><font color="Green" title="E73">A8</font></span></a>, <a class="txt" href="#E13:71"><span class="lab"><font color="Green" title="E76">A11</font></span></a>, <a class="txt" href="#E16:71"><span class="lab"><font color="Green" title="E79">A14</font></span></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></span>;<br/>
<a NAME="E21:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A18</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:71"><span class="lab"><font color="Green" title="E82">A17</font></span></a>, <a class="ref" href="anproj_1.html#T9" title="ANPROJ_1:th.9">ANPROJ_1:9</a></span>;<br/>
<a NAME="E22:71"/>
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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="txt" href="#E5:71"><span class="lab"><font color="Green" title="E69">A4</font></span></a>, <a class="txt" href="#E6:71"><span class="lab"><font color="Green" title="E70">A5</font></span></a>, <a class="txt" href="#E8:71"><span class="lab"><font color="Green" title="E72">A7</font></span></a>, <a class="txt" href="#E9:71"><span class="lab"><font color="Green" title="E73">A8</font></span></a>, <a class="txt" href="#E14:71"><span class="lab"><font color="Green" title="E77">A12</font></span></a>, <a class="txt" href="#E17:71"><span class="lab"><font color="Green" title="E80">A15</font></span></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E24:71"/><span class="lab"><font color="Green" title="E84">A19</font></span>: 
( <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>  )
 <span class="kw">and </span><br/><a NAME="E25:71"/><span class="lab"><font color="Green" title="E85">A20</font></span>: 
not <font color="Maroon" title="c13">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:71"><span class="lab"><font color="Green" title="E75">A10</font></span></a>, <a class="txt" href="#E21:71"><span class="lab"><font color="Green" title="E83">A18</font></span></a>, <a class="ref" href="anproj_1.html#T20" title="ANPROJ_1:th.20">ANPROJ_1:20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">r2</font> =  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">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#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:71"><span class="lab"><font color="Green" title="E85">A20</font></span></a>, <a class="ref" href="anproj_1.html#T42" title="ANPROJ_1:th.42">ANPROJ_1:42</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">r2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">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/>

<span class="kw">thus </span><a NAME="E27:71"/>
( <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:71"><span class="lab"><font color="Green" title="E69">A4</font></span></a>, <a class="txt" href="#E7:71"><span class="lab"><font color="Green" title="E71">A6</font></span></a>, <a class="txt" href="#E8:71"><span class="lab"><font color="Green" title="E72">A7</font></span></a>, <a class="txt" href="#E10:71"><span class="lab"><font color="Green" title="E74">A9</font></span></a>, <a class="txt" href="#E14:71"><span class="lab"><font color="Green" title="E77">A12</font></span></a>, <a class="txt" href="#E15:71"><span class="lab"><font color="Green" title="E78">A13</font></span></a>, <a class="txt" href="#E17:71"><span class="lab"><font color="Green" title="E80">A15</font></span></a>, <a class="txt" href="#E18:71"><span class="lab"><font color="Green" title="E81">A16</font></span></a>, <a class="txt" href="#E24:71"><span class="lab"><font color="Green" title="E84">A19</font></span></a>, <a class="txt" href="#E25:71"><span class="lab"><font color="Green" title="E85">A20</font></span></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
