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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>;<br/>

<b>given </b><font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">w</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><a NAME="E2:95"/><i><font color="Green" title="E85">A1</font></i>: 
( ( for <font color="Olive" title="b1">w1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">a1</font>, <font color="Olive" title="b5">b1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Olive" title="b1">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">a1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">b1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">w</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">a1</font>, <font color="Olive" title="b4">b1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">a1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">b1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> holds <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) )
 ;<br/>

<a NAME="E3:95"/><i><font color="Green" title="E86">A2</font></i>: 
( <font color="Maroon" title="c2">y</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c3">u</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c4">v</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c5">w</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; not <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">v</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#E2:95"><i><font color="Green" title="E85">A1</font></i></a>, <a class="ref" href="anproj_2.html#T2" target="_self" title="ANPROJ_2:th.2">Th2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">p</font> =  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c7">q1</font> =  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c8">q2</font> =  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c4">v</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="ref" href="anproj_1.html#T42" title="ANPROJ_1:th.42">ANPROJ_1:42</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">p</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c7">q1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c8">q2</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1"/>
<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c8">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ;<br/><a NAME="E2:95_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c8">q2</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="collsp.html#U1" title="COLLSP:sel.1">Collinearity</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="ref" href="collsp.html#D2" title="COLLSP:def.2">COLLSP:def 2</a></i>;<br/><b>hence </b><a NAME="E3:95_1"/>
contradiction
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:95"><i><font color="Green" title="E86">A2</font></i></a>, <a class="ref" href="anproj_1.html#T41" title="ANPROJ_1:th.41">ANPROJ_1:41</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:95"/>
not <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c8">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c9">r1</font>, <font color="Maroon" title="c10">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>consider </b><font color="Maroon" title="c11">u1</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="E8:95"/><i><font color="Green" title="E87">A3</font></i>: 
( <font color="Maroon" title="c11">u1</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c9">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="c11">u1</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="c12">u2</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="E10:95"/><i><font color="Green" title="E88">A4</font></i>: 
( <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="c10">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="c12">u2</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="c13">w1</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="E12:95"/><i><font color="Green" title="E89">A5</font></i>: 
( <font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">u2</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="c3">u</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c13">w1</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c14">w2</font>,<font color="Maroon" title="c13">w1</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c13">w1</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>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:95"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="anproj_2.html#E8:95"><i><font color="Green" title="E87">A3</font></i></a>, <a class="txt" href="anproj_2.html#E10:95"><i><font color="Green" title="E88">A4</font></i></a>, <a class="ref" href="anproj_2.html#T4" target="_self" title="ANPROJ_2:th.4">Th4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">q3</font> =  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c13">w1</font>, <font color="Maroon" title="c16">r3</font> =  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c14">w2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="anproj_2.html#E12:95"><i><font color="Green" title="E89">A5</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="c15">q3</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c16">r3</font>
;<br/>

<b>thus </b><a NAME="E14:95"/>
<font color="Maroon" title="c9">r1</font>,<font color="Maroon" title="c10">r2</font>,<font color="Maroon" title="c16">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E8:95"><i><font color="Green" title="E87">A3</font></i></a>, <a class="txt" href="anproj_2.html#E10:95"><i><font color="Green" title="E88">A4</font></i></a>, <a class="txt" href="anproj_2.html#E12:95"><i><font color="Green" title="E89">A5</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>

<b>thus </b><a NAME="E15:95"/>
<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c8">q2</font>,<font color="Maroon" title="c15">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:95"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="anproj_2.html#E12:95"><i><font color="Green" title="E89">A5</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>

<b>thus </b><a NAME="E16:95"/>
<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c16">r3</font>,<font color="Maroon" title="c15">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:95"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="anproj_2.html#E12:95"><i><font color="Green" title="E89">A5</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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