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<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">tfsm1</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">tfsm2</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">tfsm3</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>assume </b><a NAME="E1:82"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">tfsm1</font>,<font color="Maroon">tfsm2</font> <a href="fsm_1.html#R7" target="_self">-are_isomorphic</a>  &amp; <font color="Maroon">tfsm2</font>,<font color="Maroon">tfsm3</font> <a href="fsm_1.html#R7" target="_self">-are_isomorphic</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon">Tf1</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm2</font><b> such that </b><br/><a NAME="E3:82"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">Tf1</font> is <a href="funct_2.html#V3">bijective</a> &amp; <font color="Maroon">Tf1</font> <a href="funct_2.html#K3">.</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">tfsm1</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">tfsm2</font> &amp; ( for <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">tfsm1</font><br/> for <font color="Olive">s1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font> holds <br/> ( <font color="Maroon">Tf1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">q</font>,<font color="Olive">s1</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm2</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">Tf1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span>,<font color="Olive">s1</font> &amp; the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">tfsm1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">q</font>,<font color="Olive">s1</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">tfsm2</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">Tf1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span>,<font color="Olive">s1</font> ) ) )
 <b>by </b><i><a class="ref" href="fsm_1.html#T11" target="_self">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon">Tf2</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm3</font><b> such that </b><br/><a NAME="E5:82"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">Tf2</font> is <a href="funct_2.html#V3">bijective</a> &amp; <font color="Maroon">Tf2</font> <a href="funct_2.html#K3">.</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">tfsm2</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">tfsm3</font> &amp; ( for <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">tfsm2</font><br/> for <font color="Olive">s1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font> holds <br/> ( <font color="Maroon">Tf2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm2</font> <a href="binop_1.html#K2">.</a> <font color="Olive">q</font>,<font color="Olive">s1</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm3</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">Tf2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span>,<font color="Olive">s1</font> &amp; the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">tfsm2</font> <a href="binop_1.html#K2">.</a> <font color="Olive">q</font>,<font color="Olive">s1</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">tfsm3</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">Tf2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span>,<font color="Olive">s1</font> ) ) )
 <b>by </b><i>, <a class="ref" href="fsm_1.html#T11" target="_self">Th11</a></i>;<br/>
<b>take </b><font color="Maroon">Tf</font> = <font color="Maroon">Tf2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">Tf1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="fsm_1.html#D19" target="_self">FSM_1:def 19</a><br/>

<b>thus </b><a NAME="E6:82"/>
<font color="Maroon">Tf</font> is <a href="funct_2.html#V3">bijective</a>
 <b>by </b><i><a class="ref" href="fsm_1.html#T12" target="_self">Th12</a>, , </i>;<br/>

<a NAME="E7:82"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">Tf1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:82"/>
<font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">tfsm1</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">tfsm3</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T12" target="_self">Th12</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">tfsm1</font>;<br/><b>let </b><font color="Maroon">s1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font>;<br/><b>thus </b>the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm3</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">s1</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">tfsm3</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">Tf2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">Tf1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span></span>)</span>,<font color="Maroon">s1</font>
<b>by </b><i><a class="ref" href="fsm_1.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">Tf2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm2</font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">Tf1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">s1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">Tf2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">Tf1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">q</font>,<font color="Maroon">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fsm_1.html#T12" target="_self">Th12</a></i>
<br/>.= 
<font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">q</font>,<font color="Maroon">s1</font></span>)</span>
<b>by </b><i><a class="ref" href="fsm_1.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>thus </b>the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">tfsm3</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">s1</font></span><a href="domain_1.html#K1">]</a></span> = 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">tfsm3</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">Tf2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">Tf1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span></span>)</span>,<font color="Maroon">s1</font>
<b>by </b><i><a class="ref" href="fsm_1.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">tfsm2</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">Tf1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">s1</font>
<b>by </b><i/>
<br/>.= 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">tfsm1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">q</font>,<font color="Maroon">s1</font>
<b>by </b><i><a class="ref" href="fsm_1.html#T12" target="_self">Th12</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:82"/>
for <font color="Olive">q11</font> being  <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">tfsm1</font><br/> for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font> holds <br/> ( <font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">q11</font>,<font color="Olive">s</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm3</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q11</font></span>)</span>,<font color="Olive">s</font> &amp; the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">tfsm1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">q11</font>,<font color="Olive">s</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">tfsm3</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q11</font></span>)</span>,<font color="Olive">s</font> )
 ;<br/>


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