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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">EqR1</font>, <font color="Olive" title="b2">EqR2</font>, <font color="Olive" title="b3">E</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">EqR1</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Olive" title="b2">EqR2</font> holds <br/> for <font color="Olive" title="b4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> or  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Olive" title="b3">E</font>,<font color="Olive" title="b4">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Olive" title="b1">EqR1</font>,<font color="Olive" title="b4">x</font>) or  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Olive" title="b3">E</font>,<font color="Olive" title="b4">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Olive" title="b2">EqR2</font>,<font color="Olive" title="b4">x</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">EqR1</font>, <font color="Maroon" title="c3">EqR2</font>, <font color="Maroon" title="c4">E</font> be   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c3">EqR2</font> implies  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> or  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">EqR1</font>,<font color="Olive" title="b1">x</font>) or  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">EqR2</font>,<font color="Olive" title="b1">x</font>) ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:36"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c4">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c3">EqR2</font>
 ; <a class="txt" 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> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> or  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">EqR1</font>,<font color="Olive" title="b1">x</font>) or  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">EqR2</font>,<font color="Olive" title="b1">x</font>) )</span><br/>

<a NAME="E2:36"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> or  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">EqR1</font>,<font color="Olive" title="b1">x</font>) or  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">EqR2</font>,<font color="Olive" title="b1">x</font>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> or  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c5">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">EqR1</font>,<font color="Maroon" title="c5">x</font>) or  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c5">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">EqR2</font>,<font color="Maroon" title="c5">x</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:36_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c5">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">EqR1</font>,<font color="Maroon" title="c5">x</font>) or  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c5">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">EqR2</font>,<font color="Maroon" title="c5">x</font>) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:36_1"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
 not  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c5">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">EqR1</font>,<font color="Maroon" title="c5">x</font>)
 <span class="kw">and </span><br/><a NAME="E3:36_1"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
 not  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c5">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">EqR2</font>,<font color="Maroon" title="c5">x</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:36_1"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
( ( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c5">x</font>) &amp;  not <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">EqR1</font>,<font color="Maroon" title="c5">x</font>) ) or ( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">EqR1</font>,<font color="Maroon" title="c5">x</font>) &amp;  not <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c5">x</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E23">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:36_1_1"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">EqR1</font>,<font color="Maroon" title="c5">x</font>)
 <span class="kw">and </span><br/><a NAME="E2:36_1_1"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
 not <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c5">x</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:36_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">EqR1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="eqrel_1.html#T19" target="_self" title="EQREL_1:th.19">Th27</a></span>;<br/><a NAME="E4:36_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E5:36_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="eqrel_1.html#T19" target="_self" title="EQREL_1:th.19">Th27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:36_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A8</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="eqrel_1.html#T19" target="_self" title="EQREL_1:th.19">Th27</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:36_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
( ( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c5">x</font>) &amp;  not <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">EqR2</font>,<font color="Maroon" title="c5">x</font>) ) or ( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">EqR2</font>,<font color="Maroon" title="c5">x</font>) &amp;  not <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c5">x</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E26">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:36_1_2"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">EqR2</font>,<font color="Maroon" title="c5">x</font>)
 <span class="kw">and </span><br/><a NAME="E2:36_1_2"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
 not <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c5">x</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:36_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">EqR2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1_2"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="eqrel_1.html#T19" target="_self" title="EQREL_1:th.19">Th27</a></span>;<br/><a NAME="E4:36_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E5:36_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1_2"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="eqrel_1.html#T19" target="_self" title="EQREL_1:th.19">Th27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E11:36_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A13</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="ref" href="eqrel_1.html#T19" target="_self" title="EQREL_1:th.19">Th27</a></span>;<br/>
<a NAME="E12:36_1"/>
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">EqR2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E10:36_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="ref" href="eqrel_1.html#T19" target="_self" title="EQREL_1:th.19">Th27</a></span>;<br/>
<a NAME="E13:36_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A14</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">EqR1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E11:36_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E29">A15</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:36_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">EqR1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:36_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">EqR1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T6" target="_self" title="EQREL_1:th.6">Th12</a></span>;<br/><a NAME="E3:36_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">EqR1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:36_1"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="ref" href="eqrel_1.html#T6" target="_self" title="EQREL_1:th.6">Th12</a></span>;<br/><a NAME="E4:36_1_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">EqR1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1_3"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="eqrel_1.html#T7" target="_self" title="EQREL_1:th.7">Th13</a></span>;<br/><a NAME="E5:36_1_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">EqR1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T6" target="_self" title="EQREL_1:th.6">Th12</a></span>;<br/><span class="kw">hence </span><a NAME="E6:36_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E6:36_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="eqrel_1.html#T19" target="_self" title="EQREL_1:th.19">Th27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E15:36_1"/>
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">EqR1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E6:36_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="eqrel_1.html#T19" target="_self" title="EQREL_1:th.19">Th27</a></span>;<br/>
<a NAME="E16:36_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A17</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">EqR2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E7:36_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E31">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:36_1_4"/><span class="lab"><font color="Green" title="E32">A19</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">EqR2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:36_1_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">EqR2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:36_1"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="ref" href="eqrel_1.html#T6" target="_self" title="EQREL_1:th.6">Th12</a></span>;<br/><a NAME="E3:36_1_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">EqR2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1_4"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="eqrel_1.html#T7" target="_self" title="EQREL_1:th.7">Th13</a></span>;<br/><a NAME="E4:36_1_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">EqR2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T6" target="_self" title="EQREL_1:th.6">Th12</a></span>;<br/><span class="kw">hence </span><a NAME="E5:36_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E10:36_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="ref" href="eqrel_1.html#T19" target="_self" title="EQREL_1:th.19">Th27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E18:36_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="eqrel_1.html#T6" target="_self" title="EQREL_1:th.6">Th12</a></span>;<br/>
<a NAME="E19:36_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:36_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="ref" href="eqrel_1.html#T7" target="_self" title="EQREL_1:th.7">Th13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:36_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E17:36_1"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="txt" href="#E14:36_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:36"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> or  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">EqR1</font>,<font color="Olive" title="b1">x</font>) or  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">E</font>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">EqR2</font>,<font color="Olive" title="b1">x</font>) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
