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<span class="kw">let </span><font color="Maroon" title="c1">FCPS</font> be   <a href="anproj_2.html#NV10" title="ANPROJ_2:NV.10">up-3-dimensional</a>  <a href="anproj_2.html#NM1" title="ANPROJ_2:NM.1">CollProjectiveSpace</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">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">o</font>, <font color="Olive" title="b5">a9</font>, <font color="Olive" title="b6">b9</font>, <font color="Olive" title="b7">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FCPS</font>  st not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">o</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Olive" title="b4">o</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">a9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b4">o</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b6">b9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b4">o</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b7">c9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">a9</font> &amp; <font color="Olive" title="b4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">b9</font> &amp; <font color="Olive" title="b4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">c9</font> holds <br/>( not <font color="Olive" title="b5">a9</font>,<font color="Olive" title="b6">b9</font>,<font color="Olive" title="b7">c9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b5">a9</font>,<font color="Olive" title="b6">b9</font>,<font color="Olive" title="b7">c9</font>,<font color="Olive" title="b4">o</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">o</font>, <font color="Maroon" title="c6">a9</font>, <font color="Maroon" title="c7">b9</font>, <font color="Maroon" title="c8">c9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FCPS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">o</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">a9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">b9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c8">c9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a9</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b9</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c9</font> implies ( not <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font>,<font color="Maroon" title="c5">o</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  ) )</span><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
( not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">o</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">a9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <span class="kw">and </span><br/><a NAME="E2:30"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">b9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E3:30"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c8">c9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E4:30"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a9</font>
 <span class="kw">and </span><br/><a NAME="E5:30"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b9</font>
 <span class="kw">and </span><br/><a NAME="E6:30"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font>,<font color="Maroon" title="c5">o</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  )</span><br/>

<a NAME="E7:30"/>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/>
<a NAME="E8:30"/><span class="kw">then </span>
not <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">a9</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="projdes1.html#T19" target="_self" title="PROJDES1:th.19">Th19</a></span>;<br/>
<a NAME="E9:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A7</font></span>: 
not <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/>
<a NAME="E10:30"/>
<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c9</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="#E3:30"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/>
<a NAME="E11:30"/><span class="kw">then </span>
not <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">c9</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E9:30"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="projdes1.html#T19" target="_self" title="PROJDES1:th.19">Th19</a></span>;<br/>
<a NAME="E12:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A8</font></span>: 
not <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c8">c9</font>,<font color="Maroon" title="c3">b</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/>
<a NAME="E13:30"/>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b9</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="#E2:30"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/>
<a NAME="E14:30"/><span class="kw">then </span>
not <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c8">c9</font>,<font color="Maroon" title="c7">b9</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E12:30"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="projdes1.html#T19" target="_self" title="PROJDES1:th.19">Th19</a></span>;<br/>
<a NAME="E15:30"/><span class="kw">then </span>
not <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font>,<font color="Maroon" title="c5">o</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:30"/>
( not <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font>,<font color="Maroon" title="c5">o</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T10" target="_self" title="PROJDES1:th.10">Th10</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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