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<b>let </b><font color="Maroon" title="c1">PCPP</font> be   <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">a1</font>, <font color="Olive" title="b2">a2</font>, <font color="Olive" title="b3">b1</font>, <font color="Olive" title="b4">b2</font>, <font color="Olive" title="b5">x</font>, <font color="Olive" title="b6">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PCPP</font>  st <font color="Olive" title="b1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a2</font> &amp; <font color="Olive" title="b3">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">b2</font> &amp; <font color="Olive" title="b3">b1</font>,<font color="Olive" title="b4">b2</font>,<font color="Olive" title="b5">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">b1</font>,<font color="Olive" title="b4">b2</font>,<font color="Olive" title="b6">y</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font>,<font color="Olive" title="b5">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font>,<font color="Olive" title="b6">y</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font>,<font color="Olive" title="b3">b1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/><font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">y</font></span><br/><b>let </b><font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c3">a2</font>, <font color="Maroon" title="c4">b1</font>, <font color="Maroon" title="c5">b2</font>, <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PCPP</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">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a2</font> &amp; <font color="Maroon" title="c4">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</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">b1</font>,<font color="Maroon" title="c5">b2</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">a1</font>,<font color="Maroon" title="c3">a2</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="c2">a1</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c7">y</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )</span><br/>













<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E12">A1</font></i>: 
( <font color="Maroon" title="c2">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a2</font> &amp; <font color="Maroon" title="c4">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</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">b1</font>,<font color="Maroon" title="c5">b2</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">a1</font>,<font color="Maroon" title="c3">a2</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="c2">a1</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c7">y</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font></span><br/>

<b>assume </b><a NAME="E2:14"/><i><font color="Green" title="E13">A2</font></i>: 
not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:14"/><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c3">a2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c4">b1</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/>
<a NAME="E4:14"/><b>then </b><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">b1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:14"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="anproj_2.html#D8" title="ANPROJ_2:def.8">ANPROJ_2:def 8</a></i>;<br/>
<a NAME="E5:14"/><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c2">a1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:14"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="hessenbe.html#E3:14"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="anproj_2.html#D8" title="ANPROJ_2:def.8">ANPROJ_2:def 8</a></i>;<br/>
<a NAME="E6:14"/>
<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c3">a2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:14"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="hessenbe.html#E3:14"><i><font color="Green" title="E14">A3</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="E7:14"/>
contradiction
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:14"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="hessenbe.html#E2:14"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="hessenbe.html#E4:14"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="hessenbe.html#E5:14"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="anproj_2.html#D8" title="ANPROJ_2:def.8">ANPROJ_2:def 8</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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