<?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"> not for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <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> holds  <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="b1">x</font>,<font color="Olive" title="b3">z</font></span><br/>

<span class="kw">consider </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> 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="E2:36"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
not <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">t</font>
 <span class="kw">and </span><br/><a NAME="E3:36"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
not <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">t</font>,<font color="Maroon" title="c4">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/>
<a NAME="E4:36"/><span class="lab"><font color="Green" title="E23">A3</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">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="diraf.html#T7" target="_self" title="DIRAF:th.7">Th7</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:36_1"/><span class="lab"><font color="Green" title="E24">A4</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="c2">x</font>,<font color="Maroon" title="c4">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <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="c5">t</font></span><br/><span class="kw">thus </span><a NAME="E2:36_1"/>
not <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="c5">t</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:36_1_1"/><span class="lab"><font color="Green" title="E25">A5</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="c2">x</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"> contradiction</span><br/>

<a NAME="E2:36_1_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="c2">x</font>,<font color="Maroon" title="c5">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="diraf.html#T28" target="_self" title="DIRAF:th.28">Th28</a></span>;<br/>
<a NAME="E3:36_1_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#T26" target="_self" title="DIRAF:th.26">Th26</a></span>;<br/>
<a NAME="E4:36_1_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="ref" href="diraf.html#T27" target="_self" title="DIRAF:th.27">Th27</a></span>;<br/>
<a NAME="E5:36_1_1"/><span class="kw">then </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> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="diraf.html#T28" target="_self" title="DIRAF:th.28">Th28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:36_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E1:36_1_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="diraf.html#D4" target="_self" title="DIRAF:def.4">Def4</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></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:36"/>
not for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <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> holds  <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="b1">x</font>,<font color="Olive" title="b3">z</font>
 ; <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>
