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

<b>let </b><font color="Maroon">IAlph</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">OAlph</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">tfsm</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="fsm_1.html#L2" target="_self">Mealy-FSM</a> of <font color="Maroon">IAlph</font>,<font color="Maroon">OAlph</font>;<br/><b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>







<b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon">S</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font>;<br/>
<b>set </b><font color="Maroon">CEk</font> =  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">Ek</font> = <font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>;<br/>
<b>set </b><font color="Maroon">CEk1</font> =  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">Ek1</font> = <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>;<br/>
<b>assume </b><a NAME="E2:63"/>
 <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>
 ;<br/>

<a NAME="E3:63"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 
<b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b> <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>;<br/>
<a NAME="E4:63"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E5:63"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">CEkm</font> =  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">Ekm</font> = <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>;<br/>
<b>set </b><font color="Maroon">CEkm1</font> =  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">Ekm1</font> = <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>;<br/>
<b>assume </b><a NAME="E1:63_1"/>
 <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>
 ;<br/>

<a NAME="E2:63_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:63_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> ;<br/><a NAME="E3:63_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x'</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font>
 ;<br/><b>consider </b><font color="Maroon">qa</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:63_1_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">qa</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">qa</font> )
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">qa</font> = <font color="Maroon">qa</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> <b>by </b><i><a class="ref" href="fsm_1.html#T10" target="_self">Th10</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:63_1_1_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/><b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> <b>by </b><i/>;<br/><a NAME="E3:63_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">qa</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/><a NAME="E4:63_1_1_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">y'</font>,<font color="Maroon">qa</font>
 <b>by </b><i/>;<br/><a NAME="E5:63_1_1_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">y'</font>,<font color="Maroon">qa</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:63_1"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E6:63_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">qa</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:63_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">qa</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:63_1_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">qa</font>
 ;<br/><b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> ;<br/><a NAME="E4:63_1_1_1_1"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">qa</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/><a NAME="E5:63_1_1_1_1"/><i><font color="Green">E47</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:63_1_1_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">y'</font>,<font color="Maroon">qa</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E7:63_1_1_1_1"/><i><font color="Green">E81</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E8:63_1_1_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
1 <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">y'</font>,<font color="Maroon">qa</font>
 <b>by </b><i>, <a class="txt" href="fsm_1.html#E1:63_1_1_1"><i><font color="Green">E40</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font>;<br/><b>let </b><font color="Maroon">k1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:63_1_1_1_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><b>set </b><font color="Maroon">Sy'</font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>;<br/><b>set </b><font color="Maroon">Sqa</font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">qa</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>;<br/><a NAME="E2:63_1_1_1_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:63_1_1_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E4:63_1_1_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>,the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">qa</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="fsm_1.html#T11" target="_self">Th11</a>, , , <a class="txt" href="fsm_1.html#E1:63_1_1_1"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E5:63_1_1_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">qa</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:63_1_1_1_1_2"/>
<font color="Maroon">k1</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>,the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">qa</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E5:63_1_1_1_1"><i><font color="Green">E47</font></i></a>, , </i>;<br/></div><b>end;</b></div><a NAME="E10:63_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">y'</font>,<font color="Maroon">qa</font>
 <b>by </b><i>, , <a class="txt" href="fsm_1.html#E1:63_1_1_1"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E11:63_1_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">qa</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E12:63_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T10" target="_self">Th10</a>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/></div><b>end;</b></div><a NAME="E8:63_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">qa</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E9:63_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:63_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> ;<br/><b>consider </b><font color="Maroon">qa</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:63_1_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">qa</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">qa</font> )
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">qa</font> = <font color="Maroon">qa</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:63_1_1_2_1"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/><b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> <b>by </b><i/>;<br/><a NAME="E3:63_1_1_2_1"/><i><font color="Green">E181</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">qa</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 <b>by </b><i>, <a class="ref" href="fsm_1.html#T10" target="_self">Th10</a>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/><a NAME="E4:63_1_1_2_1"/><i><font color="Green">E182</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:63_1_1_2_1"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">y'</font>,<font color="Maroon">qa</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E6:63_1_1_2_1"/><i><font color="Green">E184</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E7:63_1_1_2_1"/><b>then </b><i><font color="Green">E185</font></i>: 
1 <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">y'</font>,<font color="Maroon">qa</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font>;<br/><b>let </b><font color="Maroon">k1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:63_1_1_2_1_1"/><i><font color="Green">E287</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><b>set </b><font color="Maroon">Sy'</font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>;<br/><b>set </b><font color="Maroon">Sqa</font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">qa</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>;<br/><a NAME="E2:63_1_1_2_1_1"/><i><font color="Green">E288</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:63_1_1_2_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E4:63_1_1_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>,the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">qa</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, , , </i>;<br/><a NAME="E5:63_1_1_2_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">qa</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:63_1_1_2_1_1"/>
<font color="Maroon">k1</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>,the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">qa</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><a NAME="E9:63_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">y'</font>,<font color="Maroon">qa</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E10:63_1_1_2_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">qa</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E11:63_1_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">qa</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:63_1_1_2"/><i><font color="Green">E289</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">qa</font>
 ;<br/><b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> ;<br/><a NAME="E4:63_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">qa</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T10" target="_self">Th10</a>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/><a NAME="E5:63_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">y'</font>,<font color="Maroon">qa</font>
 <b>by </b><i/>;<br/><a NAME="E6:63_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">y'</font>,<font color="Maroon">qa</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:63_1"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E7:63_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">qa</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:63_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/></div><b>end;</b></div><a NAME="E8:63_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">qa</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E9:63_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:63_1"/>
 <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E6:63"/>
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );</i><br/>


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