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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Rtfsm</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>   <a href="fsm_1.html#V5" title="FSM_1:attr.5">reduced</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> holds  <font color="Olive" title="b1">Rtfsm</font>, <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Olive" title="b1">Rtfsm</font> <a href="fsm_1.html#R7" title="FSM_1:pred.7">-are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">Rtfsm</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>   <a href="fsm_1.html#V5" title="FSM_1:attr.5">reduced</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">Rtfsm</font>, <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font> <a href="fsm_1.html#R7" title="FSM_1:pred.7">-are_isomorphic</a> </span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">m</font> = <font color="Maroon" title="c3">Rtfsm</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">rm</font> =  <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">fpm</font> =  <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">Rtfsm</font>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">Rtfsm</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">Rtfsm</font> = <span class="p1">(<span class="default"><a href="eqrel_1.html#K13" title="EQREL_1:func.13">proj</a> <span class="p2">(<span class="default"><a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">Tf</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Rtfsm</font>,<span class="p1">(<span class="default"><a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:76"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
 for <font color="Olive" title="b1">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">Rtfsm</font> holds  <font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">q</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span>
<div><span class="lab"><font color="Green" title="E56">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:76_1"/>
 not <font color="Maroon" title="c7">Tf</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">q1</font>, <font color="Maroon" title="c9">q2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:76_1"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
( <font color="Maroon" title="c8">q1</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">Rtfsm</font> &amp; <font color="Maroon" title="c9">q2</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">Rtfsm</font> )
 <span class="kw">and </span><br/><a NAME="E4:76_1"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
<font color="Maroon" title="c7">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">q2</font>
 <span class="kw">and </span><br/><a NAME="E5:76_1"/><span class="lab"><font color="Green" title="E59">A5</font></span>: 
<font color="Maroon" title="c8">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">FUNCT_2:19</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">q1</font> = <font color="Maroon" title="c8">q1</font>, <font color="Maroon" title="c11">q2</font> = <font color="Maroon" title="c9">q2</font> as   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c3">Rtfsm</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:76_1"><span class="lab"><font color="Green" title="E57">A3</font></span></a></span>;<br/><a NAME="E7:76_1"/>
<font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K13" title="EQREL_1:func.13">proj</a> <span class="p2">(<span class="default"><a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76"><span class="lab"><font color="Green" title="E55">A1</font></span></a></span>;<br/><a NAME="E8:76_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A6</font></span>: 
<font color="Maroon" title="c10">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D9" title="EQREL_1:def.9">EQREL_1:def 9</a></span>;<br/><a NAME="E9:76_1"/><span class="lab"><font color="Green" title="E61">A7</font></span>: 
 <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">Rtfsm</font> is  <a href="fsm_1.html#V4" title="FSM_1:attr.4">final</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D15" target="_self" title="FSM_1:def.15">Def15</a></span>;<br/><a NAME="E10:76_1"/>
<font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K13" title="EQREL_1:func.13">proj</a> <span class="p2">(<span class="default"><a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76"><span class="lab"><font color="Green" title="E55">A1</font></span></a></span>;<br/><a NAME="E11:76_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A8</font></span>: 
<font color="Maroon" title="c11">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D9" title="EQREL_1:def.9">EQREL_1:def 9</a></span>;<br/><a NAME="E12:76_1"/>
 not <font color="Maroon" title="c10">q1</font>,<font color="Maroon" title="c11">q2</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:76_1"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="ref" href="fsm_1.html#D20" target="_self" title="FSM_1:def.20">Def20</a></span>;<br/><span class="kw">hence </span><a NAME="E13:76_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:76_1"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="txt" href="#E9:76_1"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="txt" href="#E8:76_1"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="txt" href="#E11:76_1"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="ref" href="fsm_1.html#D14" target="_self" title="FSM_1:def.14">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c8">Im1</font> =  the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">Rtfsm</font>;<br/>
<a NAME="E4:76"/><span class="lab"><font color="Green" title="E57">A9</font></span>: 
 <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">Rtfsm</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">Tf</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">Rtfsm</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">Tf</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:76_2"/><span class="lab"><font color="Green" title="E58">A10</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">Rtfsm</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">Tf</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">pq</font> = <font color="Maroon" title="c9">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">Rtfsm</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">Rtfsm</font><span class="kw"> such that </span><br/><a NAME="E4:76_2"/><span class="lab"><font color="Green" title="E59">A11</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">pq</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76_2"><span class="lab"><font color="Green" title="E58">A10</font></span></a>, <a class="ref" href="finseq_4.html#T87" title="FINSEQ_4:th.87">FINSEQ_4:87</a></span>;<br/>
<a NAME="E5:76_2"/>
<font color="Maroon" title="c10">pq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K13" title="EQREL_1:func.13">proj</a> <span class="p2">(<span class="default"><a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76_2"><span class="lab"><font color="Green" title="E58">A10</font></span></a>, <a class="txt" href="#E4:76_2"><span class="lab"><font color="Green" title="E59">A11</font></span></a>, <a class="ref" href="eqrel_1.html#T65" title="EQREL_1:th.65">EQREL_1:65</a></span>;<br/>
<a NAME="E6:76_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">pq</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76"><span class="lab"><font color="Green" title="E55">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:76_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">Tf</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:76"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">Tf</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">Rtfsm</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E6:76"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">Tf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">Rtfsm</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:76"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E7:76"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A12</font></span>: 
<font color="Maroon" title="c7">Tf</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E8:76"/><span class="lab"><font color="Green" title="E59">A13</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">Rtfsm</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D18" target="_self" title="FSM_1:def.18">Def18</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E60">A14</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">q</font> be   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c3">Rtfsm</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">IAlph</font> holds <br/> ( <font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"> the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">Rtfsm</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">q</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font></span>)</span>,<font color="Olive" title="b1">s</font>) &amp;  the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">Rtfsm</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">q</font>,<font color="Olive" title="b1">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font></span>)</span>,<font color="Olive" title="b1">s</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c10">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">IAlph</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"> the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">Rtfsm</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c10">s</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font></span>)</span>,<font color="Maroon" title="c10">s</font>) &amp;  the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">Rtfsm</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c10">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font></span>)</span>,<font color="Maroon" title="c10">s</font>) )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c11">Tfq</font> = <font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font> as   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D18" target="_self" title="FSM_1:def.18">Def18</a></span>;<br/><a NAME="E2:76_3"/><span class="lab"><font color="Green" title="E61">A15</font></span>: 
 the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span> <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="c11">Tfq</font>,<font color="Maroon" title="c10">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span>
 ;<br/><a NAME="E3:76_3"/>
<font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K13" title="EQREL_1:func.13">proj</a> <span class="p2">(<span class="default"><a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76"><span class="lab"><font color="Green" title="E55">A1</font></span></a></span>;<br/><a NAME="E4:76_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A16</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D9" title="EQREL_1:def.9">EQREL_1:def 9</a></span>;<br/><a NAME="E5:76_3"/><span class="kw">then </span>
 the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">Rtfsm</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c10">s</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font></span>)</span>,<font color="Maroon" title="c10">s</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:76"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="ref" href="fsm_1.html#D18" target="_self" title="FSM_1:def.18">Def18</a></span>;<br/><a NAME="E6:76_3"/><span class="kw">then </span>
 the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font></span>)</span>,<font color="Maroon" title="c10">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K13" title="EQREL_1:func.13">proj</a> <span class="p2">(<span class="default"><a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"> the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">Rtfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c10">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:76"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E2:76_3"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="ref" href="eqrel_1.html#T65" title="EQREL_1:th.65">EQREL_1:65</a></span>;<br/><span class="kw">hence </span><a NAME="E7:76_3"/>
<font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"> the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">Rtfsm</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c10">s</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font></span>)</span>,<font color="Maroon" title="c10">s</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76"><span class="lab"><font color="Green" title="E55">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">Rtfsm</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c10">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font></span>)</span>,<font color="Maroon" title="c10">s</font>)</span><br/><span class="kw">thus </span><a NAME="E8:76_3"/>
 the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">Rtfsm</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c10">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font></span>)</span>,<font color="Maroon" title="c10">s</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:76"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E4:76_3"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="ref" href="fsm_1.html#D18" target="_self" title="FSM_1:def.18">Def18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E10:76"/>
 the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">Rtfsm</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D18" target="_self" title="FSM_1:def.18">Def18</a></span>;<br/>
<a NAME="E11:76"/><span class="kw">then </span>
 the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K13" title="EQREL_1:func.13">proj</a> <span class="p2">(<span class="default"><a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">Rtfsm</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:76"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="ref" href="eqrel_1.html#T65" title="EQREL_1:th.65">EQREL_1:65</a></span>;<br/>
<a NAME="E12:76"/><span class="kw">then </span>
<font color="Maroon" title="c7">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">Rtfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76"><span class="lab"><font color="Green" title="E55">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:76"/>
<font color="Maroon" title="c3">Rtfsm</font>, <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Rtfsm</font> <a href="fsm_1.html#R7" title="FSM_1:pred.7">-are_isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:76"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E3:76"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E7:76"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="txt" href="#E9:76"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="fsm_1.html#D19" target="_self" title="FSM_1:def.19">Def19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
