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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</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="Olive" title="b1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">x2</font> &amp; ( for <font color="Olive" title="b3">r1</font>, <font color="Olive" title="b4">r2</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>  ex <font color="Olive" title="b5">q</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="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">r1</font>,<font color="Olive" title="b4">r2</font>,<font color="Olive" title="b5">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  ) ) ) implies for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">q</font>, <font color="Olive" title="b4">q1</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>  ex <font color="Olive" title="b5">r</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="Olive" title="b1">p</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b5">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">q</font>,<font color="Olive" title="b4">q1</font>,<font color="Olive" title="b5">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  ) )</span><br/>

<b>given </b><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</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><a NAME="E2:90"/><i><font color="Green" title="E79">A1</font></i>: 
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x2</font>
 <b>and </b><br/><a NAME="E3:90"/><i><font color="Green" title="E80">A2</font></i>: 
for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</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>  ex <font color="Olive" title="b3">q</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="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Olive" title="b3">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font>,<font color="Olive" title="b3">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">q</font>, <font color="Olive" title="b4">q1</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>  ex <font color="Olive" title="b5">r</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="Olive" title="b1">p</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b5">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">q</font>,<font color="Olive" title="b4">q1</font>,<font color="Olive" title="b5">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/>

<b>let </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">p1</font>, <font color="Maroon" title="c6">q</font>, <font color="Maroon" title="c7">q1</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/>







<b>consider </b><font color="Maroon" title="c8">p3</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="E5:90"/><i><font color="Green" title="E81">A3</font></i>: 
( <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c8">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c8">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E3:90"><i><font color="Green" title="E80">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">q3</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="E7:90"/><i><font color="Green" title="E82">A4</font></i>: 
( <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c9">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c9">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E3:90"><i><font color="Green" title="E80">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">s</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="E9:90"/><i><font color="Green" title="E83">A5</font></i>: 
( <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c10">s</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c10">s</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E3:90"><i><font color="Green" title="E80">A2</font></i></a></i>;<br/>
<a NAME="E10:90"/>
( <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c9">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="#E2:90"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="#E5:90"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="#E7:90"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="#E9:90"><i><font color="Green" title="E83">A5</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</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="c11">s1</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="E12:90"/><i><font color="Green" title="E84">A6</font></i>: 
( <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c11">s1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c11">s1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D9" target="_self" title="ANPROJ_2:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">s2</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="E14:90"/><i><font color="Green" title="E85">A7</font></i>: 
( <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c12">s2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c12">s2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E3:90"><i><font color="Green" title="E80">A2</font></i></a></i>;<br/>
<a NAME="E15:90"/>
( <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c12">s2</font>,<font color="Maroon" title="c9">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c12">s2</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="#E2:90"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="#E5:90"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="#E7:90"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="#E14:90"><i><font color="Green" title="E85">A7</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</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="c13">s3</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="E17:90"/><i><font color="Green" title="E86">A8</font></i>: 
( <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c13">s3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c13">s3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D9" target="_self" title="ANPROJ_2:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">s4</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="E19:90"/><i><font color="Green" title="E87">A9</font></i>: 
( <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c14">s4</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c14">s4</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E3:90"><i><font color="Green" title="E80">A2</font></i></a></i>;<br/>
<a NAME="E20:90"/>
( <font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c14">s4</font>,<font color="Maroon" title="c8">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c14">s4</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c4">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="#E2:90"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="#E5:90"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="#E7:90"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="#E19:90"><i><font color="Green" title="E87">A9</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</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="c15">s5</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="E22:90"/><i><font color="Green" title="E88">A10</font></i>: 
( <font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c15">s5</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c15">s5</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D9" target="_self" title="ANPROJ_2:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">s6</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="E24:90"/><i><font color="Green" title="E89">A11</font></i>: 
( <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c16">s6</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c16">s6</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E3:90"><i><font color="Green" title="E80">A2</font></i></a></i>;<br/>
<a NAME="E25:90"/>
( <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c16">s6</font>,<font color="Maroon" title="c9">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c16">s6</font>,<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="#E2:90"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="#E5:90"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="#E7:90"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="#E24:90"><i><font color="Green" title="E89">A11</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</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="c17">s7</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="E27:90"/><i><font color="Green" title="E90">A12</font></i>: 
( <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c17">s7</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c17">s7</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D9" target="_self" title="ANPROJ_2:def.9">Def9</a></i>;<br/>
<div><i><font color="Green" title="E91">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:90_1"/><i><font color="Green" title="E92">A14</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> or <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><div><i><font color="Green" title="E93">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:90_1_1"/><i><font color="Green" title="E94">A16</font></i>: 
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p3</font>, <font color="Olive" title="b2">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b2">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b2">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><b>take </b><font color="Maroon" title="c18">p3</font> = <font color="Maroon" title="c8">p3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><a NAME="E2:90_1_1"/>
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c18">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="#E1:90_1_1"><i><font color="Green" title="E94">A16</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E3:90_1_1"/>
 ex <font color="Olive" title="b1">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E5:90"><i><font color="Green" title="E81">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:90_1_2"/><i><font color="Green" title="E94">A17</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">q3</font>, <font color="Olive" title="b2">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b2">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b2">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><b>take </b><font color="Maroon" title="c18">q3</font> = <font color="Maroon" title="c9">q3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><a NAME="E2:90_1_2"/>
<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c18">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="#E1:90_1_2"><i><font color="Green" title="E94">A17</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E3:90_1_2"/>
 ex <font color="Olive" title="b1">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E7:90"><i><font color="Green" title="E82">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E4:90_1"/>
 ex <font color="Olive" title="b1">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E1:90_1"><i><font color="Green" title="E92">A14</font></i></a>, <a class="txt" href="#E2:90_1"><i><font color="Green" title="E93">A15</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:90_2"/><i><font color="Green" title="E92">A18</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p1</font> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">q1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><div><i><font color="Green" title="E93">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:90_2_1"/><i><font color="Green" title="E94">A20</font></i>: 
( <font color="Maroon" title="c8">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c9">q3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">s3</font>, <font color="Olive" title="b2">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b2">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b2">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><b>take </b><font color="Maroon" title="c18">s3</font> = <font color="Maroon" title="c13">s3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><a NAME="E2:90_2_1"/>
( <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c4">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c18">s3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E5:90"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="#E17:90"><i><font color="Green" title="E86">A8</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><a NAME="E3:90_2_1"/><b>then </b><i><font color="Green" title="E95">A21</font></i>: 
<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c18">s3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="#E1:90_2_1"><i><font color="Green" title="E94">A20</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/><a NAME="E4:90_2_1"/>
( <font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c6">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c18">s3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E7:90"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="#E17:90"><i><font color="Green" title="E86">A8</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><a NAME="E5:90_2_1"/><b>then </b>
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c18">s3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="#E1:90_2_1"><i><font color="Green" title="E94">A20</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E6:90_2_1"/>
 ex <font color="Olive" title="b1">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E3:90_2_1"><i><font color="Green" title="E95">A21</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><i><font color="Green" title="E94">A22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:90_2_2"/><i><font color="Green" title="E95">A23</font></i>: 
( <font color="Maroon" title="c8">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c9">q3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">q1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">s5</font>, <font color="Olive" title="b2">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b2">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b2">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><b>take </b><font color="Maroon" title="c18">s5</font> = <font color="Maroon" title="c15">s5</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><a NAME="E2:90_2_2"/>
( <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c4">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c18">s5</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E5:90"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="#E22:90"><i><font color="Green" title="E88">A10</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><a NAME="E3:90_2_2"/><b>then </b><i><font color="Green" title="E96">A24</font></i>: 
<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c18">s5</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="#E1:90_2_2"><i><font color="Green" title="E95">A23</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/><a NAME="E4:90_2_2"/>
( <font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c6">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c18">s5</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E7:90"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="#E22:90"><i><font color="Green" title="E88">A10</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><a NAME="E5:90_2_2"/><b>then </b>
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c18">s5</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="#E1:90_2_2"><i><font color="Green" title="E95">A23</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E6:90_2_2"/>
 ex <font color="Olive" title="b1">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E3:90_2_2"><i><font color="Green" title="E96">A24</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><i><font color="Green" title="E95">A25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:90_2_3"/><i><font color="Green" title="E96">A26</font></i>: 
( <font color="Maroon" title="c8">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p1</font> &amp; <font color="Maroon" title="c9">q3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">s7</font>, <font color="Olive" title="b2">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b2">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b2">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><b>take </b><font color="Maroon" title="c18">s7</font> = <font color="Maroon" title="c17">s7</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><a NAME="E2:90_2_3"/>
( <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c4">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c5">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c18">s7</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E5:90"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="#E27:90"><i><font color="Green" title="E90">A12</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><a NAME="E3:90_2_3"/><b>then </b><i><font color="Green" title="E97">A27</font></i>: 
<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c18">s7</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="#E1:90_2_3"><i><font color="Green" title="E96">A26</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/><a NAME="E4:90_2_3"/>
( <font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c6">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c18">s7</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E7:90"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="#E27:90"><i><font color="Green" title="E90">A12</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><a NAME="E5:90_2_3"/><b>then </b>
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c18">s7</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="#E1:90_2_3"><i><font color="Green" title="E96">A26</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E6:90_2_3"/>
 ex <font color="Olive" title="b1">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E3:90_2_3"><i><font color="Green" title="E97">A27</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:90_2_4"/><i><font color="Green" title="E96">A28</font></i>: 
( <font color="Maroon" title="c8">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p1</font> &amp; <font color="Maroon" title="c9">q3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">q1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b2">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b2">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><b>take </b><font color="Maroon" title="c18">s1</font> = <font color="Maroon" title="c11">s1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/><a NAME="E2:90_2_4"/>
( <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c4">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c5">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c18">s1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E5:90"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="#E12:90"><i><font color="Green" title="E84">A6</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><a NAME="E3:90_2_4"/><b>then </b><i><font color="Green" title="E97">A29</font></i>: 
<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c18">s1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="#E1:90_2_4"><i><font color="Green" title="E96">A28</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/><a NAME="E4:90_2_4"/>
( <font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c6">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c18">s1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E7:90"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="#E12:90"><i><font color="Green" title="E84">A6</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></i>;<br/><a NAME="E5:90_2_4"/><b>then </b>
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c18">s1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="#E1:90_2_4"><i><font color="Green" title="E96">A28</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E6:90_2_4"/>
 ex <font color="Olive" title="b1">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E3:90_2_4"><i><font color="Green" title="E97">A29</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E6:90_2"/>
 ex <font color="Olive" title="b1">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E1:90_2"><i><font color="Green" title="E92">A18</font></i></a>, <a class="txt" href="#E2:90_2"><i><font color="Green" title="E93">A19</font></i></a>, <a class="txt" href="#E3:90_2"><i><font color="Green" title="E94">A22</font></i></a>, <a class="txt" href="#E4:90_2"><i><font color="Green" title="E95">A25</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E30:90"/>
 ex <font color="Olive" title="b1">r</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="c4">p</font>,<font color="Maroon" title="c5">p1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Olive" title="b1">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="#E28:90"><i><font color="Green" title="E91">A13</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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