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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">p</font>, <font color="Olive" title="b5">q</font>, <font color="Olive" title="b6">r</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">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <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">p</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <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="b5">q</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <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="b6">r</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <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="b7">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  holds <br/><font color="Olive" title="b4">p</font>,<font color="Olive" title="b5">q</font>,<font color="Olive" title="b6">r</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">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</font>, <font color="Maroon" title="c7">r</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="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> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <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">p</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <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="c6">q</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <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="c7">r</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <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="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  implies <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</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="E1:17"/><span class="lab"><font color="Green" title="E11">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> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E2:17"/><span class="lab"><font color="Green" title="E12">A2</font></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">p</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">and </span><br/><a NAME="E3:17"/><span class="lab"><font color="Green" title="E13">A3</font></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="c6">q</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">and </span><br/><a NAME="E4:17"/><span class="lab"><font color="Green" title="E14">A4</font></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="c7">r</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">and </span><br/><a NAME="E5:17"/><span class="lab"><font color="Green" title="E15">A5</font></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="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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><br/>

<a NAME="E6:17"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">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="#E1:17"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/>
<a NAME="E7:17"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">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="#E1:17"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/>
<a NAME="E8:17"/><span class="lab"><font color="Green" title="E18">A8</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">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="#E1:17"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/>
<a NAME="E9:17"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">q</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="#E1:17"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E20">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:17_1"/><span class="lab"><font color="Green" title="E21">A11</font></span>: 
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">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="#E7:17"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><span class="kw">assume </span><a NAME="E2:17_1"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">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"> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><br/><a NAME="E3:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A13</font></span>: 
not <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="E4:17_1"/><span class="lab"><font color="Green" title="E24">A14</font></span>: 
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">p</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="#E6:17"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><a NAME="E5:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A15</font></span>: 
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">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="#E3:17_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E1:17_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/><a NAME="E6:17_1"/><span class="lab"><font color="Green" title="E26">A16</font></span>: 
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="#E9:17"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><a NAME="E7:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A17</font></span>: 
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</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="#E3:17_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E4:17_1"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E28">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_1_1"/>
not <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</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"> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><br/><a NAME="E2:17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A19</font></span>: 
not <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">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="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E3:17_1_1"/>
( <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">r</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</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="#E5:17_1"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="txt" href="#E7:17_1"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><a NAME="E4:17_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">r</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="#E2:17_1_1"><span class="lab"><font color="Green" title="E29">A19</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:17_1_1"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</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#T11" target="_self" title="PROJDES1:th.11">Th11</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><a NAME="E9:17_1"/><span class="lab"><font color="Green" title="E29">A20</font></span>: 
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">r</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="#E3:17_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E1:17_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E6:17_1"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E30">A21</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_1_2"/>
not <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">r</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"> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><br/><a NAME="E2:17_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A22</font></span>: 
not <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c2">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="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E3:17_1_2"/>
( <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c2">a</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="#E5:17_1"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="txt" href="#E9:17_1"><span class="lab"><font color="Green" title="E29">A20</font></span></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><a NAME="E4:17_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c5">p</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="#E2:17_1_2"><span class="lab"><font color="Green" title="E31">A22</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:17_1_2"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</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#T11" target="_self" title="PROJDES1:th.11">Th11</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><a NAME="E11:17_1"/><span class="lab"><font color="Green" title="E31">A23</font></span>: 
<font color="Maroon" title="c6">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_1_3"/>
( <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</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="c2">a</font>,<font color="Maroon" title="c7">r</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"> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><br/><a NAME="E2:17_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">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="#E11:17_1"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="ref" href="collsp.html#T11" title="COLLSP:th.11">COLLSP:11</a></span>;<br/><a NAME="E3:17_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</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/><span class="kw">hence </span><a NAME="E4:17_1_3"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</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#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/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E13:17_1"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</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="#E8:17_1"><span class="lab"><font color="Green" title="E28">A18</font></span></a>, <a class="txt" href="#E10:17_1"><span class="lab"><font color="Green" title="E30">A21</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>
<a NAME="E11:17"/><span class="lab"><font color="Green" title="E21">A24</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">r</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="#E1:17"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E22">A25</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:17_2"/><span class="lab"><font color="Green" title="E23">A26</font></span>: 
<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">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="#E7:17"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><span class="kw">assume </span><a NAME="E2:17_2"/><span class="lab"><font color="Green" title="E24">A27</font></span>: 
not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">r</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"> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><br/><a NAME="E3:17_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A28</font></span>: 
not <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="E4:17_2"/><span class="lab"><font color="Green" title="E26">A29</font></span>: 
<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">p</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="#E8:17"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><a NAME="E5:17_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A30</font></span>: 
<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">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="#E3:17_2"><span class="lab"><font color="Green" title="E25">A28</font></span></a>, <a class="txt" href="#E1:17_2"><span class="lab"><font color="Green" title="E23">A26</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/><a NAME="E6:17_2"/><span class="lab"><font color="Green" title="E28">A31</font></span>: 
<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="#E11:17"><span class="lab"><font color="Green" title="E21">A24</font></span></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><a NAME="E7:17_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A32</font></span>: 
<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</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="#E3:17_2"><span class="lab"><font color="Green" title="E25">A28</font></span></a>, <a class="txt" href="#E4:17_2"><span class="lab"><font color="Green" title="E26">A29</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E30">A33</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_2_1"/>
not <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</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"> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><br/><a NAME="E2:17_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A34</font></span>: 
not <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">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="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E3:17_2_1"/>
( <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">q</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</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="#E5:17_2"><span class="lab"><font color="Green" title="E27">A30</font></span></a>, <a class="txt" href="#E7:17_2"><span class="lab"><font color="Green" title="E29">A32</font></span></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><a NAME="E4:17_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</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="#E2:17_2_1"><span class="lab"><font color="Green" title="E31">A34</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:17_2_1"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</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#T11" target="_self" title="PROJDES1:th.11">Th11</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><a NAME="E9:17_2"/><span class="lab"><font color="Green" title="E31">A35</font></span>: 
<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">q</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="#E3:17_2"><span class="lab"><font color="Green" title="E25">A28</font></span></a>, <a class="txt" href="#E1:17_2"><span class="lab"><font color="Green" title="E23">A26</font></span></a>, <a class="txt" href="#E6:17_2"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E32">A36</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_2_2"/>
not <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">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"> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><br/><a NAME="E2:17_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A37</font></span>: 
not <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">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="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E3:17_2_2"/>
( <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</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="#E5:17_2"><span class="lab"><font color="Green" title="E27">A30</font></span></a>, <a class="txt" href="#E9:17_2"><span class="lab"><font color="Green" title="E31">A35</font></span></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><a NAME="E4:17_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</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="#E2:17_2_2"><span class="lab"><font color="Green" title="E33">A37</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:17_2_2"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</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#T11" target="_self" title="PROJDES1:th.11">Th11</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><a NAME="E11:17_2"/><span class="lab"><font color="Green" title="E33">A38</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_2"><span class="lab"><font color="Green" title="E24">A27</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_2_3"/>
( <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">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"> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><br/><a NAME="E2:17_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c5">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="#E11:17_2"><span class="lab"><font color="Green" title="E33">A38</font></span></a>, <a class="ref" href="collsp.html#T11" title="COLLSP:th.11">COLLSP:11</a></span>;<br/><a NAME="E3:17_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</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/><span class="kw">hence </span><a NAME="E4:17_2_3"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</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#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/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E13:17_2"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</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="#E8:17_2"><span class="lab"><font color="Green" title="E30">A33</font></span></a>, <a class="txt" href="#E10:17_2"><span class="lab"><font color="Green" title="E32">A36</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>
<a NAME="E13:17"/><span class="lab"><font color="Green" title="E23">A39</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">p</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="#E1:17"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E24">A40</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><span class="kw">now </span></a><div class="add"><a NAME="E1:17_3"/><span class="lab"><font color="Green" title="E25">A41</font></span>: 
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">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="#E8:17"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><span class="kw">assume </span><a NAME="E2:17_3"/><span class="lab"><font color="Green" title="E26">A42</font></span>: 
not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">p</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"> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><br/><a NAME="E3:17_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A43</font></span>: 
not <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="E4:17_3"/><span class="lab"><font color="Green" title="E28">A44</font></span>: 
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">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="#E6:17"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><a NAME="E5:17_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A45</font></span>: 
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">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="#E3:17_3"><span class="lab"><font color="Green" title="E27">A43</font></span></a>, <a class="txt" href="#E1:17_3"><span class="lab"><font color="Green" title="E25">A41</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/><a NAME="E6:17_3"/><span class="lab"><font color="Green" title="E30">A46</font></span>: 
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="#E13:17"><span class="lab"><font color="Green" title="E23">A39</font></span></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><a NAME="E7:17_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A47</font></span>: 
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">q</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="#E3:17_3"><span class="lab"><font color="Green" title="E27">A43</font></span></a>, <a class="txt" href="#E4:17_3"><span class="lab"><font color="Green" title="E28">A44</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E32">A48</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_3_1"/>
not <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">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"> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><br/><a NAME="E2:17_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A49</font></span>: 
not <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">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="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E3:17_3_1"/>
( <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">r</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</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="#E5:17_3"><span class="lab"><font color="Green" title="E29">A45</font></span></a>, <a class="txt" href="#E7:17_3"><span class="lab"><font color="Green" title="E31">A47</font></span></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><span class="kw">hence </span><a NAME="E4:17_3_1"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</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="#E2:17_3_1"><span class="lab"><font color="Green" title="E33">A49</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</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><a NAME="E9:17_3"/><span class="lab"><font color="Green" title="E33">A50</font></span>: 
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">r</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="#E3:17_3"><span class="lab"><font color="Green" title="E27">A43</font></span></a>, <a class="txt" href="#E1:17_3"><span class="lab"><font color="Green" title="E25">A41</font></span></a>, <a class="txt" href="#E6:17_3"><span class="lab"><font color="Green" title="E30">A46</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E34">A51</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_3_2"/>
not <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">r</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"> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><br/><a NAME="E2:17_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A52</font></span>: 
not <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c2">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="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E3:17_3_2"/>
( <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">q</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c2">a</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="#E5:17_3"><span class="lab"><font color="Green" title="E29">A45</font></span></a>, <a class="txt" href="#E9:17_3"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><a NAME="E4:17_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c6">q</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="#E2:17_3_2"><span class="lab"><font color="Green" title="E35">A52</font></span></a>, <a class="txt" href="projdes1.html#E16"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:17_3_2"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</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#T11" target="_self" title="PROJDES1:th.11">Th11</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><a NAME="E11:17_3"/><span class="lab"><font color="Green" title="E35">A53</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_3"><span class="lab"><font color="Green" title="E26">A42</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_3_3"/>
( <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</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="c5">p</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">r</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"> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><br/><a NAME="E2:17_3_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">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="#E11:17_3"><span class="lab"><font color="Green" title="E35">A53</font></span></a>, <a class="ref" href="collsp.html#T11" title="COLLSP:th.11">COLLSP:11</a></span>;<br/><span class="kw">hence </span><a NAME="E3:17_3_3"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</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#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/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E13:17_3"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</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="#E8:17_3"><span class="lab"><font color="Green" title="E32">A48</font></span></a>, <a class="txt" href="#E10:17_3"><span class="lab"><font color="Green" title="E34">A51</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>
<a NAME="E15:17"/><span class="lab"><font color="Green" title="E25">A54</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_4"/>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">r</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"> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><br/><a NAME="E2:17_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">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="#E15:17"><span class="lab"><font color="Green" title="E25">A54</font></span></a>, <a class="ref" href="anproj_2.html#D8" title="ANPROJ_2:def.8">ANPROJ_2:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:17_4"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</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#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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E17:17"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</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="#E14:17"><span class="lab"><font color="Green" title="E24">A40</font></span></a>, <a class="txt" href="#E10:17"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E12:17"><span class="lab"><font color="Green" title="E22">A25</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>
