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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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> or <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> or <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> or <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">d</font> or <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">d</font> or <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> ) holds <br/><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> </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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> or <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> or <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> or <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> ) implies <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>  )</span><br/>









<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> or <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> or <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> or <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</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> </span><br/>

<div><i><font color="Green" title="E18">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_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> or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> or <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</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> </span><br/><a NAME="E2:21_1"/><b>then </b>
<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> 
 <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="E3:21_1"/>
<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> 
 <b>by </b><i><a class="ref" href="projdes1.html#T10" target="_self" title="PROJDES1:th.10">Th10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E19">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_2"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> or <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</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> </span><br/><a NAME="E2:21_2"/><b>then </b>
<font color="Maroon" title="c5">d</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> 
 <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="E3:21_2"/>
<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> 
 <b>by </b><i><a class="ref" href="projdes1.html#T10" target="_self" title="PROJDES1:th.10">Th10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_3"/>
<font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</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> </span><br/><a NAME="E2:21_3"/><b>then </b>
<font color="Maroon" title="c3">b</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="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></i>;<br/><b>hence </b><a NAME="E3:21_3"/>
<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> 
 <b>by </b><i><a class="ref" href="projdes1.html#T10" target="_self" title="PROJDES1:th.10">Th10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:21"/>
<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> 
 <b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E2:21"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E3:21"><i><font color="Green" title="E19">A3</font></i></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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