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<b>let </b><font color="Maroon">V</font> be  non <a href="realset2.html#V3">trivial</a> <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/>

<b>given </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">v</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">w</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><a NAME="E1:95"/><i><font color="Green">E58</font></i>: 
( ( for <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">a1</font>, <font color="Olive">b1</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">w1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">a1</font>, <font color="Olive">b1</font> being  <a href="real_1.html#NM1">Real</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> holds <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">a1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> 0 ) ) )
 ;<br/>

<a NAME="E2:95"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">y</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">u</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">v</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">w</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; not <font color="Maroon">y</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> =  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">y</font>, <font color="Maroon">q1</font> =  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">u</font>, <font color="Maroon">q2</font> =  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">v</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> <b>by </b><i><a class="ref" href="anproj_1.html#T42">ANPROJ_1:42</a></i>;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

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

<b>take </b>
<font color="Maroon">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">p</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><a NAME="E2:95_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="domain_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="collsp.html#U1">Collinearity</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="collsp.html#D2">COLLSP:def 2</a></i>;<br/><b>hence </b><a NAME="E3:95_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="anproj_1.html#T41">ANPROJ_1:41</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:95"/>
not <font color="Maroon">p</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/>

<b>let </b><font color="Maroon">r1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">r2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/>



<b>consider </b><font color="Maroon">u1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E7:95"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">u1</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">u1</font> )
 <b>by </b><i><a class="ref" href="anproj_1.html#T42">ANPROJ_1:42</a></i>;<br/>
<b>consider </b><font color="Maroon">u2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E9:95"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">u2</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">u2</font> )
 <b>by </b><i><a class="ref" href="anproj_1.html#T42">ANPROJ_1:42</a></i>;<br/>
<b>consider </b><font color="Maroon">w1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">w2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E11:95"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">u1</font>,<font color="Maroon">u2</font>,<font color="Maroon">w2</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">w1</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">y</font>,<font color="Maroon">w2</font>,<font color="Maroon">w1</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">w1</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">w2</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 <b>by </b><i>, , , </i>;<br/>
<b>reconsider </b><font color="Maroon">q3</font> =  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">w1</font>, <font color="Maroon">r3</font> =  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">w2</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> <b>by </b><i>, <a class="ref" href="anproj_1.html#T42">ANPROJ_1:42</a></i>;<br/>
<b>take </b>
<font color="Maroon">q3</font>
;<br/>

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

<b>thus </b><a NAME="E13:95"/>
<font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , , <a class="txt" href="anproj_2.html#E9:95"><i><font color="Green">E61</font></i></a></i>;<br/>

<b>thus </b><a NAME="E14:95"/>
<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , <a class="txt" href="anproj_2.html#E9:95"><i><font color="Green">E61</font></i></a></i>;<br/>

<b>thus </b><a NAME="E15:95"/>
<font color="Maroon">p</font>,<font color="Maroon">r3</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , <a class="txt" href="anproj_2.html#E9:95"><i><font color="Green">E61</font></i></a></i>;<br/>


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