<?xml version="1.0"?>
<div class="add">

<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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font>, <font color="Olive" title="b4">t</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> &amp; <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>  &amp; <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b3">z</font>,<font color="Olive" title="b4">t</font> holds <br/><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b4">t</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font>, <font color="Maroon" title="c5">t</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:()"><span class="comment"><font color="Red">::  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> &amp; <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</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> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">t</font> implies <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">t</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:49"/><span class="lab"><font color="Green" title="E29">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>
 <span class="kw">and </span><br/><a NAME="E2:49"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E3:49"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">t</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="49_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:49_1"/>
<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="diraf.html#D5" target="_self" title="DIRAF:def.5">Def5</a></span>;<br/><a NAME="E2:49_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="diraf.html#T28" target="_self" title="DIRAF:th.28">Th28</a></span>;<br/><a NAME="E3:49_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c2">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T27" target="_self" title="DIRAF:th.27">Th27</a></span>;<br/><a NAME="E4:49_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">t</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D5" target="_self" title="DIRAF:def.5">Def5</a></span>;<br/><a NAME="E5:49_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">t</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T36" target="_self" title="DIRAF:th.36">Th36</a></span>;<br/><span class="kw">assume </span><a NAME="E6:49_1"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<font color="Maroon" title="c4">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">t</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a> </span><br/><a NAME="E7:49_1"/>
( <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c3">y</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="c4">z</font>,<font color="Maroon" title="c2">x</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="diraf.html#T36" target="_self" title="DIRAF:th.36">Th36</a>, <a class="ref" href="diraf.html#T37" target="_self" title="DIRAF:th.37">Th37</a></span>;<br/><span class="kw">hence </span><a NAME="E8:49_1"/>
<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">t</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_1"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E5:49_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="diraf.html#T38" target="_self" title="DIRAF:th.38">Th38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:49"/>
<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">t</font> <a href="diraf.html#NR4" title="DIRAF:NR.4">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="diraf.html#D5" target="_self" title="DIRAF:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
