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

<b>set </b><font color="Maroon" title="c5">S</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font>;<br/>
<b>let </b><font color="Maroon" title="c6">IT1</font>, <font color="Maroon" title="c7">IT2</font> be   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font>;<br/>

<b>assume </b><a NAME="E1:60_1_2"/><i><font color="Green" title="E41">A9</font></i>: 
for <font color="Olive" title="b1">qa</font>, <font color="Olive" title="b2">qb</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">qa</font>,<font color="Olive" title="b2">qb</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">IT1</font> iff <font color="Maroon" title="c4">i</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Olive" title="b1">qa</font>,<font color="Olive" title="b2">qb</font> )
 ;<br/>

<b>assume </b><a NAME="E2:60_1_2"/><i><font color="Green" title="E42">A10</font></i>: 
for <font color="Olive" title="b1">qa</font>, <font color="Olive" title="b2">qb</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">qa</font>,<font color="Olive" title="b2">qb</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">IT2</font> iff <font color="Maroon" title="c4">i</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Olive" title="b1">qa</font>,<font color="Olive" title="b2">qb</font> )
 ;<br/>

<b>assume </b><a NAME="E3:60_1_2"/>
<font color="Maroon" title="c6">IT1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">IT2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:60_1_2"/><i><font color="Green" title="E43">A11</font></i>: 
( ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">IT1</font> &amp; not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">IT2</font> ) or ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">IT2</font> &amp; not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">IT1</font> ) )
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_1_2_1"/>
( ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">IT1</font> &amp; not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">IT2</font> ) or ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">IT2</font> &amp; not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">IT1</font> ) )
 </span><b>by </b><i><a class="txt" href="fsm_1.html#E5:60_1_2"><i><font color="Green" title="E43">A11</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_1"><b>suppose </b></a><a NAME="E1:60_1_2_1_1"/><i><font color="Green" title="E44">A12</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">IT1</font> &amp; not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">IT2</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">qa</font>, <font color="Maroon" title="c10">qb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:60_1_2_1_1"/><i><font color="Green" title="E45">A13</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c10">qb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">qa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> &amp; <font color="Maroon" title="c10">qb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">qa</font> = <font color="Maroon" title="c9">qa</font>, <font color="Maroon" title="c12">qb</font> = <font color="Maroon" title="c10">qb</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> <b>by </b><i><a class="txt" href="fsm_1.html#E3:60_1_2_1_1"><i><font color="Green" title="E45">A13</font></i></a></i>;<br/><a NAME="E5:60_1_2_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c11">qa</font>,<font color="Maroon" title="c12">qb</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:60_1_2"><i><font color="Green" title="E41">A9</font></i></a>, <a class="txt" href="fsm_1.html#E1:60_1_2_1_1"><i><font color="Green" title="E44">A12</font></i></a>, <a class="txt" href="fsm_1.html#E3:60_1_2_1_1"><i><font color="Green" title="E45">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E6:60_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:60_1_2"><i><font color="Green" title="E42">A10</font></i></a>, <a class="txt" href="fsm_1.html#E1:60_1_2_1_1"><i><font color="Green" title="E44">A12</font></i></a>, <a class="txt" href="fsm_1.html#E3:60_1_2_1_1"><i><font color="Green" title="E45">A13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_2"><b>suppose </b></a><a NAME="E1:60_1_2_1_2"/><i><font color="Green" title="E44">A14</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">IT2</font> &amp; not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">IT1</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">qa</font>, <font color="Maroon" title="c10">qb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:60_1_2_1_2"/><i><font color="Green" title="E45">A15</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c10">qb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">qa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> &amp; <font color="Maroon" title="c10">qb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">qa</font> = <font color="Maroon" title="c9">qa</font>, <font color="Maroon" title="c12">qb</font> = <font color="Maroon" title="c10">qb</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> <b>by </b><i><a class="txt" href="fsm_1.html#E3:60_1_2_1_2"><i><font color="Green" title="E45">A15</font></i></a></i>;<br/><a NAME="E5:60_1_2_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c11">qa</font>,<font color="Maroon" title="c12">qb</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:60_1_2"><i><font color="Green" title="E42">A10</font></i></a>, <a class="txt" href="fsm_1.html#E1:60_1_2_1_2"><i><font color="Green" title="E44">A14</font></i></a>, <a class="txt" href="fsm_1.html#E3:60_1_2_1_2"><i><font color="Green" title="E45">A15</font></i></a></i>;<br/><b>hence </b><a NAME="E6:60_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:60_1_2"><i><font color="Green" title="E41">A9</font></i></a>, <a class="txt" href="fsm_1.html#E1:60_1_2_1_2"><i><font color="Green" title="E44">A14</font></i></a>, <a class="txt" href="fsm_1.html#E3:60_1_2_1_2"><i><font color="Green" title="E45">A15</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
