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<b>let </b><font color="Maroon" title="c1">FCPS</font> be  <a href="anproj_2.html#NV10" title="ANPROJ_2:NV.10">up-3-dimensional</a> <a href="anproj_2.html#NM1" title="ANPROJ_2:NM.1">CollProjectiveSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FCPS</font>  st <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">d</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  holds <br/><font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>,<font color="Olive" title="b1">a</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><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">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FCPS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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">d</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  implies <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c2">a</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a>  )</span><br/>









<b>assume </b><a NAME="E1:10"/>
<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">d</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c2">a</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> </span><br/>

<b>then consider </b><font color="Maroon" title="c6">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="E3:10"/><i><font color="Green" title="E8">A1</font></i>: 
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">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">d</font>,<font color="Maroon" title="c6">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/>
<a NAME="E4:10"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c2">a</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E3:10"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></i>;<br/>
<a NAME="E5:10"/>
<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E3:10"><i><font color="Green" title="E8">A1</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="c7">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="E7:10"/><i><font color="Green" title="E10">A3</font></i>: 
( <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">y</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="c5">d</font>,<font color="Maroon" title="c7">y</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E4:10"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="anproj_2.html#D9" title="ANPROJ_2:def.9">ANPROJ_2:def 9</a></i>;<br/>
<a NAME="E8:10"/>
( <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">y</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">y</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="projdes1.html#E7:10"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E9:10"/>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c2">a</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <b>by </b><i><a class="ref" href="projdes1.html#D1" target="_self" title="PROJDES1:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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