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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font>, <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font>, <font color="Olive" title="b6">c</font>, <font color="Olive" title="b7">s</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">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">b</font>,<font color="Olive" title="b6">c</font>,<font color="Olive" title="b1">p</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">b</font>,<font color="Olive" title="b6">c</font>,<font color="Olive" title="b3">r</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">b</font>,<font color="Olive" title="b6">c</font>,<font color="Olive" title="b2">q</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font>,<font color="Olive" title="b7">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  holds <br/><font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">b</font>,<font color="Olive" title="b6">c</font>,<font color="Olive" title="b7">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">r</font>, <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">s</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c2">p</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c4">r</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c3">q</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  implies <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  )</span><br/>















<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E12">A1</font></i>: 
( not <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c2">p</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c4">r</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c3">q</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1"/><i><font color="Green" title="E13">A2</font></i>: 
not <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</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"> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><br/><a NAME="E2:15_1"/><b>then </b><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E1:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projdes1.html#E18"><i><font color="Green" title="E11">Lm5</font></i></a></i>;<br/><a NAME="E3:15_1"/><i><font color="Green" title="E15">A4</font></i>: 
not <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E1:15_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></i>;<br/><a NAME="E4:15_1"/>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></i>;<br/><a NAME="E5:15_1"/><b>then </b><i><font color="Green" title="E16">A5</font></i>: 
( <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c2">p</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c3">q</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c4">r</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E1:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="projdes1.html#T10" target="_self" title="PROJDES1:th.10">Th10</a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></i>;<br/><a NAME="E6:15_1"/><i><font color="Green" title="E17">A6</font></i>: 
not <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E1:15_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></i>;<br/><a NAME="E7:15_1"/>
( <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c2">p</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c3">q</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c4">r</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E1:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></i>;<br/><a NAME="E8:15_1"/><b>then </b>
<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E6:15_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="projdes1.html#E18"><i><font color="Green" title="E11">Lm5</font></i></a></i>;<br/><a NAME="E9:15_1"/><b>then </b>
( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">a</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c6">b</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c7">c</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E2:15_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="projdes1.html#E3:15_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="projdes1.html#E5:15_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a>, <a class="ref" href="projdes1.html#T12" target="_self" title="PROJDES1:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E10:15_1"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E1:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="projdes1.html#T12" target="_self" title="PROJDES1:th.12">Th12</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="E3:15"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <b>by </b><i><a class="ref" href="projdes1.html#T10" target="_self" title="PROJDES1:th.10">Th10</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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