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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="analoaf.html#NM1" title="ANALOAF:NM.1">OAffinSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#R3" title="DIRAF:pred.3">are_collinear</a> </span><br/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#R3" title="DIRAF:pred.3">are_collinear</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:37"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#R3" title="DIRAF:pred.3">are_collinear</a> </span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:37"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
 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#R3" title="DIRAF:pred.3">are_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T36" target="_self" onmouseover="rs('diraf/T36')" onmouseout="rh()">Th36</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:37"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
 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#R3" title="DIRAF:pred.3">are_collinear</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<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#R3" title="DIRAF:pred.3">are_collinear</a> 
 
;<br/>
<a NAME="E6:37"/>
( <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#R3" title="DIRAF:pred.3">are_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#R3" title="DIRAF:pred.3">are_collinear</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E29">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:37"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E5:37"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="diraf.html#T32" target="_self" onmouseover="rs('diraf/T32')" onmouseout="rh()">Th32</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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