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<b>let </b><font color="Maroon" title="c1">S</font> be   <a href="analoaf.html#NM1" title="ANALOAF:NM.1">OAffinSpace</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">u</font>, <font color="Olive" title="b2">z</font>, <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font>, <font color="Olive" title="b5">w</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">z</font> &amp; <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font>,<font color="Olive" title="b1">u</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  &amp; <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font>,<font color="Olive" title="b2">z</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  &amp; <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">z</font>,<font color="Olive" title="b5">w</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  holds <br/><font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font>,<font color="Olive" title="b5">w</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a> </span><br/><b>let </b><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">z</font>, <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</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">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">z</font> &amp; <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">u</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  &amp; <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">z</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  &amp; <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c6">w</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  implies <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">w</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  )</span><br/>











<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c2">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">z</font> &amp; <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">u</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  &amp; <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">z</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  &amp; <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c6">w</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">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="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">w</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">w</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a> </span><br/><a NAME="E2:45_1"/>
( <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c5">y</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  &amp; <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="diraf.html#T37" target="_self" title="DIRAF:th.37">Th37</a></i>;<br/><a NAME="E3:45_1"/><b>then </b>
( <font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c5">y</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  &amp; <font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">x</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  &amp; <font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c6">w</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="diraf.html#E1:45"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="diraf.html#E1:45_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="diraf.html#T36" target="_self" title="DIRAF:th.36">Th36</a>, <a class="ref" href="diraf.html#T38" target="_self" title="DIRAF:th.38">Th38</a></i>;<br/><b>hence </b><a NAME="E4:45_1"/>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">w</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a> 
 <b>by </b><i><a class="txt" href="diraf.html#E1:45"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="diraf.html#T38" target="_self" title="DIRAF:th.38">Th38</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="E3:45"/>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">w</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a> 
 <b>by </b><i><a class="ref" href="diraf.html#T37" target="_self" title="DIRAF:th.37">Th37</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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