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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">x</font>, <font color="Olive" title="b2">y</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">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/> ex <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st not <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a> </span><br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</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">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font> implies  ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st not <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Olive" title="b1">z</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  )</span><br/>





<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not for <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Olive" title="b1">z</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a> </span><br/>

<b>assume </b><a NAME="E2:47"/><i><font color="Green" title="E31">A2</font></i>: 
for <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Olive" title="b1">z</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"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E4:47"/><i><font color="Green" title="E32">A3</font></i>: 
not <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a> 
 <b>by </b><i><a class="ref" href="diraf.html#T42" target="_self" title="DIRAF:th.42">Th42</a></i>;<br/>
<a NAME="E5:47"/>
( <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">a</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  &amp; <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">b</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  &amp; <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c6">c</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="#E2:47"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:47"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:47"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="#E4:47"><i><font color="Green" title="E32">A3</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/>


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