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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>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">d</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">b'</font>, <font color="Maroon" title="c6">d'</font>, <font color="Maroon" title="c7">q</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><a NAME="E1:96"/><i><font color="Green" title="E86">A1</font></i>: 
( not <font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">d</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c2">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c4">b</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c6">d'</font>,<font color="Maroon" title="c3">d</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ;<br/>

<a NAME="E2:96"/><b>then </b><i><font color="Green" title="E87">A2</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a></i>;<br/>
<a NAME="E3:96"/><i><font color="Green" title="E88">A3</font></i>: 
<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c4">b</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E1:96"><i><font color="Green" title="E86">A1</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/>
<div><i><font color="Green" title="E89">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:96_1"/><i><font color="Green" title="E90">A5</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b'</font>
 ;<br/><a NAME="E2:96_1"/><i><font color="Green" title="E91">A6</font></i>: 
<font color="Maroon" title="c6">d'</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c7">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:96"><i><font color="Green" title="E86">A1</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><a NAME="E3:96_1"/>
<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c2">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:96"><i><font color="Green" title="E86">A1</font></i></a>, <a class="txt" href="anproj_2.html#E1:96_1"><i><font color="Green" title="E90">A5</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">o</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">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> such that </b><br/><a NAME="E5:96_1"/><i><font color="Green" title="E92">A7</font></i>: 
( <font color="Maroon" title="c6">d'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c8">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:96_1"><i><font color="Green" title="E91">A6</font></i></a>, <a class="ref" href="anproj_2.html#D9" target="_self" title="ANPROJ_2:def.9">Def9</a></i>;<br/><a NAME="E6:96_1"/>
( <font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c6">d'</font>,<font color="Maroon" title="c8">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E5:96_1"><i><font color="Green" title="E92">A7</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><b>hence </b><a NAME="E7:96_1"/>
 ex <font color="Olive" title="b1">o</font> being  <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> st <br/>( <font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c6">d'</font>,<font color="Olive" title="b1">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:96_2"/><i><font color="Green" title="E90">A8</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b'</font>
 ;<br/><b>consider </b><font color="Maroon" title="c8">z</font> being   <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><b> such that </b><br/><a NAME="E3:96_2"/><i><font color="Green" title="E91">A9</font></i>: 
( <font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c6">d'</font>,<font color="Maroon" title="c8">z</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c8">z</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:96"><i><font color="Green" title="E86">A1</font></i></a>, <a class="txt" href="anproj_2.html#E3:96"><i><font color="Green" title="E88">A3</font></i></a>, <a class="ref" href="anproj_2.html#D9" target="_self" title="ANPROJ_2:def.9">Def9</a></i>;<br/><a NAME="E4:96_2"/><i><font color="Green" title="E92">A10</font></i>: 
<font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:96_2_1"/>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">b</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E2:96_2_1"/>
<font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:96"><i><font color="Green" title="E86">A1</font></i></a>, <a class="txt" href="anproj_2.html#E2:96"><i><font color="Green" title="E87">A2</font></i></a>, <a class="txt" href="anproj_2.html#E3:96_2"><i><font color="Green" title="E91">A9</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:96_2"/>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:96"><i><font color="Green" title="E86">A1</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">o</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">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> such that </b><br/><a NAME="E7:96_2"/><i><font color="Green" title="E93">A11</font></i>: 
( <font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c9">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c9">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:96_2"><i><font color="Green" title="E92">A10</font></i></a>, <a class="ref" href="anproj_2.html#D9" target="_self" title="ANPROJ_2:def.9">Def9</a></i>;<br/><a NAME="E8:96_2"/><i><font color="Green" title="E94">A12</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:96_2_2"/>
not <font color="Maroon" title="c8">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b'</font>
 ;<br/>

<a NAME="E2:96_2_2"/><b>then </b><i><font color="Green" title="E95">A13</font></i>: 
( <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c3">d</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E1:96"><i><font color="Green" title="E86">A1</font></i></a>, <a class="txt" href="anproj_2.html#E3:96_2"><i><font color="Green" title="E91">A9</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/>
<a NAME="E3:96_2_2"/>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c4">b</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E4:96_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:96"><i><font color="Green" title="E86">A1</font></i></a>, <a class="txt" href="anproj_2.html#E1:96_2"><i><font color="Green" title="E90">A8</font></i></a>, <a class="txt" href="anproj_2.html#E2:96_2_2"><i><font color="Green" title="E95">A13</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:96_2"/><i><font color="Green" title="E95">A14</font></i>: 
<font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c6">d'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:96_2"><i><font color="Green" title="E91">A9</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><a NAME="E10:96_2"/>
<font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c5">b'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a></i>;<br/><a NAME="E11:96_2"/><b>then </b><i><font color="Green" title="E96">A15</font></i>: 
<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c6">d'</font>,<font color="Maroon" title="c9">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E7:96_2"><i><font color="Green" title="E93">A11</font></i></a>, <a class="txt" href="anproj_2.html#E8:96_2"><i><font color="Green" title="E94">A12</font></i></a>, <a class="txt" href="anproj_2.html#E9:96_2"><i><font color="Green" title="E95">A14</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/><a NAME="E12:96_2"/>
<font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c9">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E7:96_2"><i><font color="Green" title="E93">A11</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><b>hence </b><a NAME="E13:96_2"/>
 ex <font color="Olive" title="b1">o</font> being  <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> st <br/>( <font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c6">d'</font>,<font color="Olive" title="b1">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E11:96_2"><i><font color="Green" title="E96">A15</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:96"/>
 ex <font color="Olive" title="b1">o</font> being  <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> st <br/>( <font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c6">d'</font>,<font color="Olive" title="b1">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:96"><i><font color="Green" title="E89">A4</font></i></a></i>;<br/>


</div>
