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

<b>let </b><font color="Maroon" title="c1">IAlph</font>, <font color="Maroon" title="c2">OAlph</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">tfsm</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="fsm_1.html#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Maroon" title="c1">IAlph</font>,<font color="Maroon" title="c2">OAlph</font>;<br/><b>let </b><font color="Maroon" title="c4">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>







<b>reconsider </b><font color="Maroon" title="c5">k'</font> = <font color="Maroon" title="c4">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">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>set </b><font color="Maroon" title="c7">CEk</font> =  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">Ek</font> = <font color="Maroon" title="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font>;<br/>
<b>set </b><font color="Maroon" title="c9">CEk1</font> =  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">Ek1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font>;<br/>
<b>assume </b><a NAME="E2:61"/>
 <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>
 ;<br/>

<a NAME="E3:61"/><b>then </b><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font>
 
<b>by </b><i><a class="ref" href="fsm_1.html#T8" target="_self" title="FSM_1:th.8">Th8</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b> <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1</span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>;<br/>
<a NAME="E4:61"/><i><font color="Green" title="E45">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E5:61"/><i><font color="Green" title="E46">A3</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>set </b><font color="Maroon" title="c12">CEkm</font> =  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c13">Ekm</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font>;<br/>
<b>set </b><font color="Maroon" title="c14">CEkm1</font> =  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c15">Ekm1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font>;<br/>
<b>assume </b><a NAME="E1:61_1"/>
 <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>
 ;<br/>

<a NAME="E2:61_1"/><b>then </b><i><font color="Green" title="E47">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font>
 
<b>by </b><i><a class="ref" href="fsm_1.html#T8" target="_self" title="FSM_1:th.8">Th8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:61_1_1_1"/><i><font color="Green" title="E48">A5</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c17">x'</font> = <font color="Maroon" title="c16">x</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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/><a NAME="E3:61_1_1_1"/><i><font color="Green" title="E49">A6</font></i>: 
<font color="Maroon" title="c17">x'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font>
 ;<br/><b>consider </b><font color="Maroon" title="c18">qa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:61_1_1_1"/><i><font color="Green" title="E50">A7</font></i>: 
( <font color="Maroon" title="c18">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="c17">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c18">qa</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:61_1_1_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">qa</font> = <font color="Maroon" title="c18">qa</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#E5:61_1_1_1"><i><font color="Green" title="E50">A7</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:61_1_1_1_1_1"/><i><font color="Green" title="E51">A8</font></i>: 
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c21">y'</font> = <font color="Maroon" title="c20">y</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#E3:61_1_1_1"><i><font color="Green" title="E49">A6</font></i></a></i>;<br/><a NAME="E3:61_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">y</font>,<font color="Maroon" title="c19">qa</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E5:61_1_1_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="txt" href="fsm_1.html#E1:61_1_1_1_1_1"><i><font color="Green" title="E51">A8</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/><a NAME="E4:61_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c19">qa</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D12" target="_self" title="FSM_1:def.12">Def12</a></i>;<br/><a NAME="E5:61_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">k</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c19">qa</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T43" target="_self" title="FSM_1:th.43">Th43</a></i>;<br/><a NAME="E6:61_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c19">qa</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="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D12" target="_self" title="FSM_1:def.12">Def12</a></i>;<br/><b>hence </b><a NAME="E7:61_1_1_1_1_1"/>
<font color="Maroon" title="c20">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c19">qa</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:61_1_1_1_1"/><i><font color="Green" title="E51">A9</font></i>: 
<font color="Maroon" title="c20">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c19">qa</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c21">y'</font> = <font color="Maroon" title="c20">y</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> ;<br/><a NAME="E4:61_1_1_1_1"/><i><font color="Green" title="E52">A10</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c19">qa</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="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:61_1_1_1_1"><i><font color="Green" title="E51">A9</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/><a NAME="E5:61_1_1_1_1"/><i><font color="Green" title="E53">A11</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E6:61_1_1_1_1"/><i><font color="Green" title="E54">A12</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c19">qa</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:61"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="fsm_1.html#E4:61_1_1_1_1"><i><font color="Green" title="E52">A10</font></i></a>, <a class="ref" href="fsm_1.html#D12" target="_self" title="FSM_1:def.12">Def12</a></i>;<br/><a NAME="E7:61_1_1_1_1"/><i><font color="Green" title="E55">A13</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E8:61_1_1_1_1"/><b>then </b><i><font color="Green" title="E56">A14</font></i>: 
1 <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c19">qa</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E6:61_1_1_1_1"><i><font color="Green" title="E54">A12</font></i></a>, <a class="ref" href="fsm_1.html#T44" target="_self" title="FSM_1:th.44">Th44</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">IAlph</font>;<br/><b>let </b><font color="Maroon" title="c23">k1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:61_1_1_1_1_2"/><i><font color="Green" title="E57">A15</font></i>: 
<font color="Maroon" title="c23">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><b>set </b><font color="Maroon" title="c24">Sy'</font> = the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c22">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/><b>set </b><font color="Maroon" title="c25">Sqa</font> = the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">qa</font>,<font color="Maroon" title="c22">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/><a NAME="E2:61_1_1_1_1_2"/><i><font color="Green" title="E58">A16</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:61_1_1_1_1_2"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><a NAME="E4:61_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">k</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c22">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">qa</font>,<font color="Maroon" title="c22">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:61_1_1_1_1_2"><i><font color="Green" title="E58">A16</font></i></a>, <a class="txt" href="fsm_1.html#E6:61_1_1_1_1"><i><font color="Green" title="E54">A12</font></i></a>, <a class="txt" href="fsm_1.html#E7:61_1_1_1_1"><i><font color="Green" title="E55">A13</font></i></a>, <a class="ref" href="fsm_1.html#T44" target="_self" title="FSM_1:th.44">Th44</a></i>;<br/><a NAME="E5:61_1_1_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c22">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">qa</font>,<font color="Maroon" title="c22">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></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="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D12" target="_self" title="FSM_1:def.12">Def12</a></i>;<br/><b>hence </b><a NAME="E6:61_1_1_1_1_2"/>
<font color="Maroon" title="c23">k1</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c22">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">qa</font>,<font color="Maroon" title="c22">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:61_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="fsm_1.html#E1:61_1_1_1_1_2"><i><font color="Green" title="E57">A15</font></i></a>, <a class="ref" href="fsm_1.html#D12" target="_self" title="FSM_1:def.12">Def12</a></i>;<br/></div><b>end;</b></div><a NAME="E10:61_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c19">qa</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E5:61_1_1_1_1"><i><font color="Green" title="E53">A11</font></i></a>, <a class="txt" href="fsm_1.html#E8:61_1_1_1_1"><i><font color="Green" title="E56">A14</font></i></a>, <a class="ref" href="fsm_1.html#T44" target="_self" title="FSM_1:th.44">Th44</a></i>;<br/><a NAME="E11:61_1_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c19">qa</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D12" target="_self" title="FSM_1:def.12">Def12</a></i>;<br/><b>hence </b><a NAME="E12:61_1_1_1_1"/>
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E5:61_1_1_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/></div><b>end;</b></div><a NAME="E8:61_1_1_1"/><b>then </b>
<font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c19">qa</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E9:61_1_1_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:61_1_1"/><i><font color="Green" title="E48">A17</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c17">x'</font> = <font color="Maroon" title="c16">x</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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>consider </b><font color="Maroon" title="c18">qa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:61_1_1"/><i><font color="Green" title="E49">A18</font></i>: 
( <font color="Maroon" title="c18">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="c17">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c18">qa</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:61_1_1"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">qa</font> = <font color="Maroon" title="c18">qa</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#E5:61_1_1"><i><font color="Green" title="E49">A18</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:61_1_1_2_1"/><i><font color="Green" title="E50">A19</font></i>: 
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c21">y'</font> = <font color="Maroon" title="c20">y</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#E5:61_1_1"><i><font color="Green" title="E49">A18</font></i></a></i>;<br/><a NAME="E3:61_1_1_2_1"/><i><font color="Green" title="E51">A20</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c19">qa</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="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E5:61_1_1"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="fsm_1.html#E1:61_1_1_2_1"><i><font color="Green" title="E50">A19</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/><a NAME="E4:61_1_1_2_1"/><i><font color="Green" title="E52">A21</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E5:61_1_1_2_1"/><i><font color="Green" title="E53">A22</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c19">qa</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:61"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="fsm_1.html#E3:61_1_1_2_1"><i><font color="Green" title="E51">A20</font></i></a>, <a class="ref" href="fsm_1.html#D12" target="_self" title="FSM_1:def.12">Def12</a></i>;<br/><a NAME="E6:61_1_1_2_1"/><i><font color="Green" title="E54">A23</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E7:61_1_1_2_1"/><b>then </b><i><font color="Green" title="E55">A24</font></i>: 
1 <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c19">qa</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E5:61_1_1_2_1"><i><font color="Green" title="E53">A22</font></i></a>, <a class="ref" href="fsm_1.html#T44" target="_self" title="FSM_1:th.44">Th44</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">IAlph</font>;<br/><b>let </b><font color="Maroon" title="c23">k1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:61_1_1_2_1_1"/><i><font color="Green" title="E56">A25</font></i>: 
<font color="Maroon" title="c23">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><b>set </b><font color="Maroon" title="c24">Sy'</font> = the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c22">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/><b>set </b><font color="Maroon" title="c25">Sqa</font> = the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">qa</font>,<font color="Maroon" title="c22">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/><a NAME="E2:61_1_1_2_1_1"/><i><font color="Green" title="E57">A26</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:61_1_1_2_1_1"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><a NAME="E4:61_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c4">k</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c22">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">qa</font>,<font color="Maroon" title="c22">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:61_1_1_2_1_1"><i><font color="Green" title="E57">A26</font></i></a>, <a class="txt" href="fsm_1.html#E5:61_1_1_2_1"><i><font color="Green" title="E53">A22</font></i></a>, <a class="txt" href="fsm_1.html#E6:61_1_1_2_1"><i><font color="Green" title="E54">A23</font></i></a>, <a class="ref" href="fsm_1.html#T44" target="_self" title="FSM_1:th.44">Th44</a></i>;<br/><a NAME="E5:61_1_1_2_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c22">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">qa</font>,<font color="Maroon" title="c22">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></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="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D12" target="_self" title="FSM_1:def.12">Def12</a></i>;<br/><b>hence </b><a NAME="E6:61_1_1_2_1_1"/>
<font color="Maroon" title="c23">k1</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c22">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">qa</font>,<font color="Maroon" title="c22">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:61_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="fsm_1.html#E1:61_1_1_2_1_1"><i><font color="Green" title="E56">A25</font></i></a>, <a class="ref" href="fsm_1.html#D12" target="_self" title="FSM_1:def.12">Def12</a></i>;<br/></div><b>end;</b></div><a NAME="E9:61_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c19">qa</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:61_1_1_2_1"><i><font color="Green" title="E52">A21</font></i></a>, <a class="txt" href="fsm_1.html#E7:61_1_1_2_1"><i><font color="Green" title="E55">A24</font></i></a>, <a class="ref" href="fsm_1.html#T44" target="_self" title="FSM_1:th.44">Th44</a></i>;<br/><a NAME="E10:61_1_1_2_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c19">qa</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D12" target="_self" title="FSM_1:def.12">Def12</a></i>;<br/><b>hence </b><a NAME="E11:61_1_1_2_1"/>
<font color="Maroon" title="c20">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c19">qa</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:61_1_1_2"/><i><font color="Green" title="E50">A27</font></i>: 
<font color="Maroon" title="c20">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c19">qa</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c21">y'</font> = <font color="Maroon" title="c20">y</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> ;<br/><a NAME="E4:61_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">y</font>,<font color="Maroon" title="c19">qa</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:61_1_1_2"><i><font color="Green" title="E50">A27</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/><a NAME="E5:61_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c19">qa</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D12" target="_self" title="FSM_1:def.12">Def12</a></i>;<br/><a NAME="E6:61_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">k</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c19">qa</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T43" target="_self" title="FSM_1:th.43">Th43</a></i>;<br/><a NAME="E7:61_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">y'</font>,<font color="Maroon" title="c19">qa</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="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D12" target="_self" title="FSM_1:def.12">Def12</a></i>;<br/><b>hence </b><a NAME="E8:61_1_1_2"/>
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E5:61_1_1"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/></div><b>end;</b></div><a NAME="E8:61_1_1"/><b>then </b>
<font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c19">qa</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E9:61_1_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:61_1"/>
 <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E6:61"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="fsm_1.html#E4:61"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="fsm_1.html#E5:61"><i><font color="Green" title="E46">A3</font></i></a>);</i><br/>


</div>
