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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> 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>  holds <br/><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">p</font>,<font color="Olive" title="b5">q</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> 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>  implies <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><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16"/><span class="lab"><font color="Green" title="E10">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="E2:16"/><span class="lab"><font color="Green" title="E11">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="E3:16"/><span class="lab"><font color="Green" title="E12">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> 
 ; <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="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><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FCPS</font><span class="kw"> such that </span><br/><a NAME="E5:16"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E6:16"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">x</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="#E2:16"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="projdes1.html#D1" target="_self" title="PROJDES1:def.1">Def1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FCPS</font><span class="kw"> such that </span><br/><a NAME="E8:16"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">y</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E9:16"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c8">y</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="#E3:16"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="projdes1.html#D1" target="_self" title="PROJDES1:def.1">Def1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E17">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_1"/><span class="lab"><font color="Green" title="E18">A9</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>
 ; <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="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><br/><a NAME="E2:16_1"/>
( <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">y</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E3:16_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="collsp.html#T11" title="COLLSP:th.11">COLLSP:11</a></span>;<br/><a NAME="E4:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A10</font></span>: 
<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="E5:16_1"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E1:16_1"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="collsp.html#T11" title="COLLSP:th.11">COLLSP:11</a></span>;<br/><a NAME="E6:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A11</font></span>: 
<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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/><div><span class="lab"><font color="Green" title="E21">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:16_1_1"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FCPS</font><span class="kw"> such that </span><br/><a NAME="E3:16_1_1"/><span class="lab"><font color="Green" title="E22">A13</font></span>: 
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c9">z</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E4:16_1_1"/><span class="lab"><font color="Green" title="E23">A14</font></span>: 
<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="anproj_2.html#D9" title="ANPROJ_2:def.9">ANPROJ_2:def 9</a></span>;<br/><span class="kw">assume </span><a NAME="E5:16_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font>
 ; <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="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><br/><a NAME="E6:16_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c9">z</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_1"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E4:16_1"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E23">A14</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="E7:16_1_1"/>
<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="#E3:16_1_1"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="ref" href="projdes1.html#D1" target="_self" title="PROJDES1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 ; <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="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><br/><a NAME="E2:16_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A15</font></span>: 
<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c4">c</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="#E9:16"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E3:16_1_2"/>
<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E4:16_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</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="#E1:16"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E2:16_1_2"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="ref" href="collsp.html#T11" title="COLLSP:th.11">COLLSP:11</a></span>;<br/><a NAME="E5:16_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">x</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="E6:16_1_2"/>
<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="#E5:16"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="projdes1.html#D1" target="_self" title="PROJDES1:def.1">Def1</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="E9:16_1"/>
<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="#E7:16_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_2"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ; <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="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><br/><a NAME="E2:16_2"/><span class="kw">then </span>
<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> 
 <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/><span class="kw">hence </span><a NAME="E3:16_2"/>
<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="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="E12:16"/>
<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="#E10:16"><span class="lab"><font color="Green" title="E17">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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