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<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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">V</font> is  <a href="anproj_2.html#V1" title="ANPROJ_2:attr.1">up-3-dimensional</a>  implies  <a href="anproj_1.html#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font> is  <a href="collsp.html#V4" title="COLLSP:attr.4">proper</a>  )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</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><a NAME="E2:73"/><span class="lab"><font color="Green" title="E68">A1</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</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"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<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">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="b3">c</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">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">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="anproj_2.html#D6" target="_self" title="ANPROJ_2:def.6">ANPROJ_2:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="anproj_1.html#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font> is  <a href="collsp.html#V4" title="COLLSP:attr.4">proper</a> </span><br/>

<a NAME="E3:73"/><span class="lab"><font color="Green" title="E69">A2</font></span>: 
 not <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">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="#E2:73"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="anproj_2.html#T1" target="_self" title="ANPROJ_2:th.1">Th1</a></span>;<br/>
<a NAME="E4:73"/><span class="lab"><font color="Green" title="E70">A3</font></span>: 
 not <font color="Maroon" title="c4">w</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="#E2:73"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="anproj_2.html#T1" target="_self" title="ANPROJ_2:th.1">Th1</a></span>;<br/>
<a NAME="E5:73"/><span class="lab"><font color="Green" title="E71">A4</font></span>: 
(  not <font color="Maroon" title="c2">u</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Maroon" title="c3">v</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="#E2:73"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="anproj_2.html#T1" target="_self" title="ANPROJ_2:th.1">Th1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">p</font> =  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c6">q</font> =  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c7">r</font> =  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Maroon" title="c4">w</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="#E4:73"><span class="lab"><font color="Green" title="E70">A3</font></span></a>, <a class="ref" href="anproj_1.html#T26" title="ANPROJ_1:th.26">ANPROJ_1:26</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">p</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="collsp.html#D6" title="COLLSP:def.6">COLLSP:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <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">U1</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> holds  <font color="Maroon" title="c5">p</font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">q</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  for <font color="Olive">b<sub>1</sub></font> being    <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">U1</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> holds  <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c7">r</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/>

<span class="kw">assume </span><a NAME="E7:73"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</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="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E8:73"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Maroon" title="c2">u</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Maroon" title="c3">v</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Maroon" title="c4">w</font></span>)</span></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#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="ref" href="collsp.html#D2" title="COLLSP:def.2">COLLSP:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:73"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73"><span class="lab"><font color="Green" title="E71">A4</font></span></a>, <a class="txt" href="#E4:73"><span class="lab"><font color="Green" title="E70">A3</font></span></a>, <a class="txt" href="#E3:73"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="anproj_1.html#T25" title="ANPROJ_1:th.25">ANPROJ_1:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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