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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="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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#K5" title="ANPROJ_1:func.5">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#K5" title="ANPROJ_1:func.5">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#K5" title="ANPROJ_1:func.5">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#K5" title="ANPROJ_1:func.5">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/>

<span class="kw">given </span><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#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span><span class="kw"> such that </span><a NAME="E2:92"/><span class="lab"><font color="Green" title="E79">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:92"/><span class="lab"><font color="Green" title="E80">A2</font></span>: 
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#K5" title="ANPROJ_1:func.5">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#K5" title="ANPROJ_1:func.5">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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#K5" title="ANPROJ_1:func.5">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/>

<span class="kw">let </span><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#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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/>







<span class="kw">consider </span><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#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:92"/><span class="lab"><font color="Green" title="E81">A3</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E6:92"/><span class="lab"><font color="Green" title="E82">A4</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:92"><span class="lab"><font color="Green" title="E80">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><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#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:92"/><span class="lab"><font color="Green" title="E83">A5</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E9:92"/><span class="lab"><font color="Green" title="E84">A6</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:92"><span class="lab"><font color="Green" title="E80">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">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#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:92"/><span class="lab"><font color="Green" title="E85">A7</font></span>: 
<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c10">s2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E12:92"/><span class="lab"><font color="Green" title="E86">A8</font></span>: 
<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c10">s2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:92"><span class="lab"><font color="Green" title="E80">A2</font></span></a></span>;<br/>
<a NAME="E13:92"/><span class="lab"><font color="Green" title="E87">A9</font></span>: 
<font color="Maroon" title="c10">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:92"><span class="lab"><font color="Green" title="E86">A8</font></span></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">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#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E15:92"/><span class="lab"><font color="Green" title="E88">A10</font></span>: 
<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c11">s4</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E16:92"/><span class="lab"><font color="Green" title="E89">A11</font></span>: 
<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c11">s4</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:92"><span class="lab"><font color="Green" title="E80">A2</font></span></a></span>;<br/>
<a NAME="E17:92"/><span class="lab"><font color="Green" title="E90">A12</font></span>: 
<font color="Maroon" title="c11">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:92"><span class="lab"><font color="Green" title="E89">A11</font></span></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></span>;<br/>
<a NAME="E18:92"/>
<font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c10">s2</font>,<font color="Maroon" title="c9">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="txt" href="#E5:92"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="txt" href="#E8:92"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E11:92"><span class="lab"><font color="Green" title="E85">A7</font></span></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">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#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E20:92"/><span class="lab"><font color="Green" title="E91">A13</font></span>: 
<font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c12">s3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E21:92"/><span class="lab"><font color="Green" title="E92">A14</font></span>: 
<font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c12">s3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:92"><span class="lab"><font color="Green" title="E87">A9</font></span></a>, <a class="ref" href="anproj_2.html#D9" target="_self" title="ANPROJ_2:def.9">Def9</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">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#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E23:92"/><span class="lab"><font color="Green" title="E93">A15</font></span>: 
<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c13">s</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E24:92"/><span class="lab"><font color="Green" title="E94">A16</font></span>: 
<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c13">s</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:92"><span class="lab"><font color="Green" title="E80">A2</font></span></a></span>;<br/>
<a NAME="E25:92"/>
<font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c11">s4</font>,<font color="Maroon" title="c8">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="txt" href="#E5:92"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="txt" href="#E8:92"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E15:92"><span class="lab"><font color="Green" title="E88">A10</font></span></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">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#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E27:92"/><span class="lab"><font color="Green" title="E95">A17</font></span>: 
<font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c14">s5</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E28:92"/><span class="lab"><font color="Green" title="E96">A18</font></span>: 
<font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c14">s5</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:92"><span class="lab"><font color="Green" title="E90">A12</font></span></a>, <a class="ref" href="anproj_2.html#D9" target="_self" title="ANPROJ_2:def.9">Def9</a></span>;<br/>
<a NAME="E29:92"/><span class="lab"><font color="Green" title="E97">A19</font></span>: 
<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c7">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:92"><span class="lab"><font color="Green" title="E94">A16</font></span></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">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#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E31:92"/><span class="lab"><font color="Green" title="E98">A20</font></span>: 
<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c15">s6</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E32:92"/><span class="lab"><font color="Green" title="E99">A21</font></span>: 
<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c15">s6</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:92"><span class="lab"><font color="Green" title="E80">A2</font></span></a></span>;<br/>
<a NAME="E33:92"/><span class="lab"><font color="Green" title="E100">A22</font></span>: 
<font color="Maroon" title="c15">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:92"><span class="lab"><font color="Green" title="E99">A21</font></span></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></span>;<br/>
<a NAME="E34:92"/>
<font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c15">s6</font>,<font color="Maroon" title="c9">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="txt" href="#E5:92"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="txt" href="#E8:92"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E31:92"><span class="lab"><font color="Green" title="E98">A20</font></span></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">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#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E36:92"/><span class="lab"><font color="Green" title="E101">A23</font></span>: 
<font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c16">s7</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E37:92"/><span class="lab"><font color="Green" title="E102">A24</font></span>: 
<font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c16">s7</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:92"><span class="lab"><font color="Green" title="E100">A22</font></span></a>, <a class="ref" href="anproj_2.html#D9" target="_self" title="ANPROJ_2:def.9">Def9</a></span>;<br/>
<a NAME="E38:92"/>
<font color="Maroon" title="c13">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="txt" href="#E5:92"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="txt" href="#E8:92"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E23:92"><span class="lab"><font color="Green" title="E93">A15</font></span></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">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#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E40:92"/><span class="lab"><font color="Green" title="E103">A25</font></span>: 
<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c17">s1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E41:92"/><span class="lab"><font color="Green" title="E104">A26</font></span>: 
<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c17">s1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:92"><span class="lab"><font color="Green" title="E97">A19</font></span></a>, <a class="ref" href="anproj_2.html#D9" target="_self" title="ANPROJ_2:def.9">Def9</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E105">A27</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E106">A28</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:92_1_1"/><span class="lab"><font color="Green" title="E107">A29</font></span>: 
<font color="Maroon" title="c9">q3</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c17">s1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:92"><span class="lab"><font color="Green" title="E104">A26</font></span></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:92_1_1"/><span class="lab"><font color="Green" title="E108">A30</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:92_1_1"/><span class="lab"><font color="Green" title="E109">A31</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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/><a NAME="E4:92_1_1"/>
( <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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:92"><span class="lab"><font color="Green" title="E84">A6</font></span></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></span>;<br/><a NAME="E5:92_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A32</font></span>: 
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c17">s1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:92_1_1"><span class="lab"><font color="Green" title="E109">A31</font></span></a>, <a class="txt" href="#E1:92_1_1"><span class="lab"><font color="Green" title="E107">A29</font></span></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c18">s1</font> = <font color="Maroon" title="c17">s1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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="E6:92_1_1"/><span class="lab"><font color="Green" title="E111">A33</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:92"><span class="lab"><font color="Green" title="E103">A25</font></span></a>, <a class="ref" href="anproj_2.html#T25" target="_self" title="ANPROJ_2:th.25">Th25</a></span>;<br/><a NAME="E7:92_1_1"/>
( <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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:92"><span class="lab"><font color="Green" title="E82">A4</font></span></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></span>;<br/><a NAME="E8:92_1_1"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_1_1"><span class="lab"><font color="Green" title="E108">A30</font></span></a>, <a class="txt" href="#E6:92_1_1"><span class="lab"><font color="Green" title="E111">A33</font></span></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></span>;<br/><span class="kw">hence </span><a NAME="E9:92_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#K5" title="ANPROJ_1:func.5">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:92_1_1"><span class="lab"><font color="Green" title="E110">A32</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E107">A34</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:92_1_2"/><span class="lab"><font color="Green" title="E108">A35</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:92_1_2"/><span class="lab"><font color="Green" title="E109">A36</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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/><span class="kw">take </span><font color="Maroon" title="c18">s3</font> = <font color="Maroon" title="c12">s3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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="E3:92_1_2"/>
( <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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:92"><span class="lab"><font color="Green" title="E84">A6</font></span></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></span>;<br/><a NAME="E4:92_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A37</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:92"><span class="lab"><font color="Green" title="E92">A14</font></span></a>, <a class="txt" href="#E2:92_1_2"><span class="lab"><font color="Green" title="E109">A36</font></span></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></span>;<br/><a NAME="E5:92_1_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:92"><span class="lab"><font color="Green" title="E82">A4</font></span></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></span>;<br/><a NAME="E6:92_1_2"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:92"><span class="lab"><font color="Green" title="E91">A13</font></span></a>, <a class="txt" href="#E1:92_1_2"><span class="lab"><font color="Green" title="E108">A35</font></span></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></span>;<br/><span class="kw">hence </span><a NAME="E7:92_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#K5" title="ANPROJ_1:func.5">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:92_1_2"><span class="lab"><font color="Green" title="E110">A37</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E108">A38</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:92_1_3"/><span class="lab"><font color="Green" title="E109">A39</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:92_1_3"/><span class="lab"><font color="Green" title="E110">A40</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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/><span class="kw">take </span><font color="Maroon" title="c18">s7</font> = <font color="Maroon" title="c16">s7</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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="E3:92_1_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:92"><span class="lab"><font color="Green" title="E84">A6</font></span></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></span>;<br/><a NAME="E4:92_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A41</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:92"><span class="lab"><font color="Green" title="E102">A24</font></span></a>, <a class="txt" href="#E2:92_1_3"><span class="lab"><font color="Green" title="E110">A40</font></span></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></span>;<br/><a NAME="E5:92_1_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:92"><span class="lab"><font color="Green" title="E82">A4</font></span></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></span>;<br/><a NAME="E6:92_1_3"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:92"><span class="lab"><font color="Green" title="E101">A23</font></span></a>, <a class="txt" href="#E1:92_1_3"><span class="lab"><font color="Green" title="E109">A39</font></span></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></span>;<br/><span class="kw">hence </span><a NAME="E7:92_1_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#K5" title="ANPROJ_1:func.5">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:92_1_3"><span class="lab"><font color="Green" title="E111">A41</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E109">A42</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:92_1_4"/><span class="lab"><font color="Green" title="E110">A43</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:92_1_4"/><span class="lab"><font color="Green" title="E111">A44</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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/><span class="kw">take </span><font color="Maroon" title="c18">s5</font> = <font color="Maroon" title="c14">s5</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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="E3:92_1_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:92"><span class="lab"><font color="Green" title="E84">A6</font></span></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></span>;<br/><a NAME="E4:92_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A45</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:92"><span class="lab"><font color="Green" title="E95">A17</font></span></a>, <a class="txt" href="#E2:92_1_4"><span class="lab"><font color="Green" title="E111">A44</font></span></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></span>;<br/><a NAME="E5:92_1_4"/>
( <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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:92"><span class="lab"><font color="Green" title="E82">A4</font></span></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></span>;<br/><a NAME="E6:92_1_4"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:92"><span class="lab"><font color="Green" title="E96">A18</font></span></a>, <a class="txt" href="#E1:92_1_4"><span class="lab"><font color="Green" title="E110">A43</font></span></a>, <a class="ref" href="anproj_2.html#D8" target="_self" title="ANPROJ_2:def.8">Def8</a></span>;<br/><span class="kw">hence </span><a NAME="E7:92_1_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#K5" title="ANPROJ_1:func.5">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:92_1_4"><span class="lab"><font color="Green" title="E112">A45</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E5:92_1"/>
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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/><span class="kw">hence </span><a NAME="E6:92_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#K5" title="ANPROJ_1:func.5">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_1"><span class="lab"><font color="Green" title="E107">A34</font></span></a>, <a class="txt" href="#E4:92_1"><span class="lab"><font color="Green" title="E109">A42</font></span></a>, <a class="txt" href="#E3:92_1"><span class="lab"><font color="Green" title="E108">A38</font></span></a>, <a class="txt" href="#E1:92_1"><span class="lab"><font color="Green" title="E106">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E106">A46</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:92_2_1"/><span class="lab"><font color="Green" title="E107">A47</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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/><span class="kw">take </span><font color="Maroon" title="c18">q3</font> = <font color="Maroon" title="c9">q3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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="E2:92_2_1"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_2_1"><span class="lab"><font color="Green" title="E107">A47</font></span></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:92_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#K5" title="ANPROJ_1:func.5">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E9:92"><span class="lab"><font color="Green" title="E84">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E107">A48</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:92_2_2"/><span class="lab"><font color="Green" title="E108">A49</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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/><span class="kw">take </span><font color="Maroon" title="c18">p3</font> = <font color="Maroon" title="c8">p3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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="E2:92_2_2"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_2_2"><span class="lab"><font color="Green" title="E108">A49</font></span></a>, <a class="ref" href="anproj_2.html#D7" target="_self" title="ANPROJ_2:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:92_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#K5" title="ANPROJ_1:func.5">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:92"><span class="lab"><font color="Green" title="E82">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E3:92_2"/>
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="ANPROJ_1:func.5">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/><span class="kw">hence </span><a NAME="E4:92_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#K5" title="ANPROJ_1:func.5">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_2"><span class="lab"><font color="Green" title="E107">A48</font></span></a>, <a class="txt" href="#E1:92_2"><span class="lab"><font color="Green" title="E106">A46</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E44:92"/>
 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#K5" title="ANPROJ_1:func.5">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E42:92"><span class="lab"><font color="Green" title="E105">A27</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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