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<b>let </b><font color="Maroon" title="c1">FCPS</font> be  <a href="anproj_2.html#NV10" title="ANPROJ_2:NV.10">up-3-dimensional</a> <a href="anproj_2.html#NM1" title="ANPROJ_2:NM.1">CollProjectiveSpace</a>;<br/><b>let </b><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>;<br/>











<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E10">A1</font></i>: 
( 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>  )
 ;<br/>

<b>then consider </b><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><b> such that </b><br/><a NAME="E3:12"/><i><font color="Green" title="E11">A2</font></i>: 
( <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>  &amp; <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>  )
 <b>by </b><i><a class="ref" href="projdes1.html#D1" target="_self" title="PROJDES1:def.1">Def1</a></i>;<br/>
<b>consider </b><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><b> such that </b><br/><a NAME="E5:12"/><i><font color="Green" title="E12">A3</font></i>: 
( <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>  &amp; <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>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E1:12"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="projdes1.html#D1" target="_self" title="PROJDES1:def.1">Def1</a></i>;<br/>
<div><i><font color="Green" title="E13">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ;<br/><a NAME="E2:12_1"/><b>then </b>
<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> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></i>;<br/><b>hence </b><a NAME="E3:12_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> 
 <b>by </b><i><a class="ref" href="projdes1.html#T10" target="_self" title="PROJDES1:th.10">Th10</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2"/><i><font color="Green" title="E14">A5</font></i>: 
<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>
 ;<br/><a NAME="E2:12_2"/><b>then </b><i><font color="Green" title="E15">A6</font></i>: 
<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> 
 <b>by </b><i><a class="txt" href="projdes1.html#E3:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="projdes1.html#E5:12"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="collsp.html#T11" title="COLLSP:th.11">COLLSP:11</a></i>;<br/><a NAME="E3:12_2"/>
( <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>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E3:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="projdes1.html#E5:12"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></i>;<br/><a NAME="E4:12_2"/><b>then </b>
<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> 
 <b>by </b><i><a class="txt" href="projdes1.html#E1:12_2"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="collsp.html#T11" title="COLLSP:th.11">COLLSP:11</a></i>;<br/><a NAME="E5:12_2"/><b>then </b><i><font color="Green" title="E16">A7</font></i>: 
( <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>  &amp; <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>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E2:12_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></i>;<br/><div><i><font color="Green" title="E17">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 ;<br/><a NAME="E2:12_2_1"/><b>then </b>
( <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>  &amp; <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>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E3:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="projdes1.html#E5:12"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></i>;<br/><a NAME="E3:12_2_1"/><b>then </b>
<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> 
 <b>by </b><i><a class="txt" href="projdes1.html#E1:12"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="projdes1.html#E3:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="collsp.html#T11" title="COLLSP:th.11">COLLSP:11</a></i>;<br/><a NAME="E4:12_2_1"/><b>then </b>
<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> 
 <b>by </b><i><a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></i>;<br/><b>hence </b><a NAME="E5:12_2_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> 
 <b>by </b><i><a class="txt" href="projdes1.html#E3:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="projdes1.html#D1" target="_self" title="PROJDES1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_2"/><i><font color="Green" title="E18">A9</font></i>: 
<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>
 ;<br/><a NAME="E2:12_2_2"/>
( <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>  &amp; <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>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E3:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="projdes1.html#E5:12"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></i>;<br/><b>then consider </b><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><b> such that </b><br/><a NAME="E4:12_2_2"/><i><font color="Green" title="E19">A10</font></i>: 
( <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>  &amp; <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>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D9" title="ANPROJ_2:def.9">ANPROJ_2:def 9</a></i>;<br/><a NAME="E5:12_2_2"/>
<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> 
 <b>by </b><i><a class="txt" href="projdes1.html#E5:12_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="projdes1.html#E1:12_2_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="projdes1.html#E4:12_2_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="anproj_2.html#D8" title="ANPROJ_2:def.8">ANPROJ_2:def 8</a></i>;<br/><b>hence </b><a NAME="E6:12_2_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> 
 <b>by </b><i><a class="txt" href="projdes1.html#E4:12_2_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="projdes1.html#D1" target="_self" title="PROJDES1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:12_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> 
 <b>by </b><i><a class="txt" href="projdes1.html#E6:12_2"><i><font color="Green" title="E17">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:12"/>
<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> 
 <b>by </b><i><a class="txt" href="projdes1.html#E6:12"><i><font color="Green" title="E13">A4</font></i></a></i>;<br/>


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