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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="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
 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> 
 <span class="kw">and </span><br/><a NAME="E2:15"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E3:15"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E4:15"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E5:15"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">now </span></a><div class="add"><a NAME="E1:15_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/><a NAME="E2:15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A6</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T6" target="_self" title="PROJDES1:th.6">Th10</a></span>;<br/><a NAME="E3:15_1"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="projdes1.html#T7" target="_self" title="PROJDES1:th.7">Th11</a></span>;<br/><span class="kw">assume </span><a NAME="E4:15_1"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
 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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E5:15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A9</font></span>: 
 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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E6:15_1"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E4:15_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="projdes1.html#E14"><span class="lab"><font color="Green" title="E11">Lm5</font></span></a></span>;<br/><a NAME="E7:15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T7" target="_self" title="PROJDES1:th.7">Th11</a></span>;<br/><a NAME="E8:15_1"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="projdes1.html#T7" target="_self" title="PROJDES1:th.7">Th11</a></span>;<br/><a NAME="E9:15_1"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
 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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E10:15_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="projdes1.html#T7" target="_self" title="PROJDES1:th.7">Th11</a></span>;<br/><a NAME="E11:15_1"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="txt" href="#E8:15_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="txt" href="projdes1.html#E14"><span class="lab"><font color="Green" title="E11">Lm5</font></span></a></span>;<br/><a NAME="E12:15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A13</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T7" target="_self" title="PROJDES1:th.7">Th11</a></span>;<br/><a NAME="E13:15_1"/>
( <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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="projdes1.html#T7" target="_self" title="PROJDES1:th.7">Th11</a></span>;<br/><a NAME="E14:15_1"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E2:15_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E3:15_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="projdes1.html#T8" target="_self" title="PROJDES1:th.8">Th12</a></span>;<br/><span class="kw">hence </span><a NAME="E15:15_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E7:15_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="txt" href="#E12:15_1"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="ref" href="projdes1.html#T8" target="_self" title="PROJDES1:th.8">Th12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:15"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T6" target="_self" title="PROJDES1:th.6">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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