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<b>let </b><font color="Maroon" title="c1">V</font> be  non <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a> <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">p</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">ANPROJ_2:def 8</a><br/><b>let </b><font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">r</font>, <font color="Maroon" title="c5">r1</font>, <font color="Maroon" title="c6">r2</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>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:67"/><i><font color="Green" title="E64">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font>
 <b>and </b><br/><a NAME="E2:67"/><i><font color="Green" title="E65">A2</font></i>: 
( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</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">q</font>,<font color="Maroon" title="c5">r1</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">q</font>,<font color="Maroon" title="c6">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">u</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="E4:67"/><i><font color="Green" title="E66">A3</font></i>: 
( <font color="Maroon" title="c7">u</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <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> )
 <b>by </b><i><a class="ref" href="anproj_1.html#T42" title="ANPROJ_1:th.42">ANPROJ_1:42</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">v</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="E6:67"/><i><font color="Green" title="E67">A4</font></i>: 
( <font color="Maroon" title="c8">v</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c3">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="c8">v</font> )
 <b>by </b><i><a class="ref" href="anproj_1.html#T42" title="ANPROJ_1:th.42">ANPROJ_1:42</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">u1</font>, <font color="Maroon" title="c10">v1</font>, <font color="Maroon" title="c11">w1</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="E8:67"/><i><font color="Green" title="E68">A5</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="c9">u1</font> &amp; <font color="Maroon" title="c3">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="c10">v1</font> &amp; <font color="Maroon" title="c4">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="c11">w1</font> &amp; <font color="Maroon" title="c9">u1</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c10">v1</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c11">w1</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c9">u1</font>,<font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">w1</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:67"><i><font color="Green" title="E65">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="c12">u2</font>, <font color="Maroon" title="c13">v2</font>, <font color="Maroon" title="c14">w2</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="E10:67"/><i><font color="Green" title="E69">A6</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="c12">u2</font> &amp; <font color="Maroon" title="c3">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="c13">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="c14">w2</font> &amp; <font color="Maroon" title="c12">u2</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c13">v2</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c14">w2</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c12">u2</font>,<font color="Maroon" title="c13">v2</font>,<font color="Maroon" title="c14">w2</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:67"><i><font color="Green" title="E65">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="c15">u3</font>, <font color="Maroon" title="c16">v3</font>, <font color="Maroon" title="c17">w3</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="E12:67"/><i><font color="Green" title="E70">A7</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="c15">u3</font> &amp; <font color="Maroon" title="c3">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="c16">v3</font> &amp; <font color="Maroon" title="c6">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="c17">w3</font> &amp; <font color="Maroon" title="c15">u3</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c16">v3</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c17">w3</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c15">u3</font>,<font color="Maroon" title="c16">v3</font>,<font color="Maroon" title="c17">w3</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:67"><i><font color="Green" title="E65">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="E13:67"/><i><font color="Green" title="E71">A8</font></i>: 
not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">v</font>
 
<b>by </b><i><a class="txt" href="anproj_2.html#E1:67"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="anproj_2.html#E4:67"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="anproj_2.html#E6:67"><i><font color="Green" title="E67">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="E14:67"/>
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c9">u1</font>,<font color="Maroon" title="c7">u</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c12">u2</font>,<font color="Maroon" title="c7">u</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c15">u3</font>,<font color="Maroon" title="c7">u</font> &amp;  <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="c13">v2</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="c16">v3</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">w1</font>,<font color="Maroon" title="c11">w1</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c14">w2</font>,<font color="Maroon" title="c14">w2</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c17">w3</font>,<font color="Maroon" title="c17">w3</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E4:67"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="anproj_2.html#E6:67"><i><font color="Green" title="E67">A4</font></i></a>, <a class="txt" href="anproj_2.html#E8:67"><i><font color="Green" title="E68">A5</font></i></a>, <a class="txt" href="anproj_2.html#E10:67"><i><font color="Green" title="E69">A6</font></i></a>, <a class="txt" href="anproj_2.html#E12:67"><i><font color="Green" title="E70">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="E15:67"/><b>then </b>
( <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c11">w1</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c14">w2</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c17">w3</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E8:67"><i><font color="Green" title="E68">A5</font></i></a>, <a class="txt" href="anproj_2.html#E10:67"><i><font color="Green" title="E69">A6</font></i></a>, <a class="txt" href="anproj_2.html#E12:67"><i><font color="Green" title="E70">A7</font></i></a>, <a class="ref" href="anproj_1.html#T9" title="ANPROJ_1:th.9">ANPROJ_1:9</a></i>;<br/>
<a NAME="E16:67"/><b>then </b>
<font color="Maroon" title="c11">w1</font>,<font color="Maroon" title="c14">w2</font>,<font color="Maroon" title="c17">w3</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E4:67"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="anproj_2.html#E6:67"><i><font color="Green" title="E67">A4</font></i></a>, <a class="txt" href="anproj_2.html#E13:67"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="anproj_1.html#T19" title="ANPROJ_1:th.19">ANPROJ_1:19</a></i>;<br/>
<b>hence </b><a NAME="E17:67"/>
<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c6">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E8:67"><i><font color="Green" title="E68">A5</font></i></a>, <a class="txt" href="anproj_2.html#E10:67"><i><font color="Green" title="E69">A6</font></i></a>, <a class="txt" href="anproj_2.html#E12:67"><i><font color="Green" title="E70">A7</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>


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