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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">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><b> such that </b><a NAME="E1:76"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a>  st <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</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 )
 ;<br/>

<b>let </b><font color="Maroon">p</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="anproj_2.html#D10" target="_self">ANPROJ_2:def 10</a><br/><b>let </b><font color="Maroon">q</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">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:76"/><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">p</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">y</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">w</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E5:76"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">w</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">w</font> )
 <b>by </b><i><a class="ref" href="anproj_1.html#T42">ANPROJ_1:42</a></i>;<br/>
<a NAME="E6:76"/><i><font color="Green">E61</font></i>: 
( <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>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:76"/>
not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E9:76"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">z</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">y</font>,<font color="Maroon">w</font>,<font color="Maroon">z</font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">w</font>,<font color="Maroon">z</font> )
 <b>by </b><i>, <a class="ref" href="anproj_1.html#T21">ANPROJ_1:21</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> =  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">z</font> as   <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> <b>by </b><i>, <a class="ref" href="anproj_1.html#T42">ANPROJ_1:42</a></i>;<br/>
<b>take </b>
<font color="Maroon">r</font>
;<br/>

<b>thus </b><a NAME="E11:76"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>

<b>thus </b><a NAME="E12:76"/>
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>

<b>thus </b><a NAME="E13:76"/>
<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , , <a class="txt" href="anproj_2.html#E6:76"><i><font color="Green">E61</font></i></a></i>;<br/>


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