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<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>, <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>given </b><font color="Maroon">Tf</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><a NAME="E1:81_1_2"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">Tf</font> is <a href="funct_2.html#V3">bijective</a> &amp; <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">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">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">q</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">tfsm2</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">q</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">q</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">tfsm2</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">q</font></span>)</span>,<font color="Olive">s</font> ) ) )
 ;<br/>

<a NAME="E2:81_1_2"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">Tf</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">Tf</font> is <a href="funct_2.html#V2">onto</a> )
 
<b>by </b><i><a class="ref" href="fsm_1.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E3:81_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Tf</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E4:81_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Tf</font> <a href="funct_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E5:81_1_2"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">Tf</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/>
<a NAME="E6:81_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Tf</font> <a href="funct_1.html#K2">"</a> </span>)</span> <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_1.html#T55">FUNCT_1:55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Tf'</font> = <font color="Maroon">Tf</font> <a href="funct_1.html#K2">"</a>  as   <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">tfsm1</font> <b>by </b><i><a class="ref" href="fsm_1.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/>
<b>take </b>
<font color="Maroon">Tf'</font>
;<br/>

<a NAME="E8:81_1_2"/>
( <font color="Maroon">Tf'</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">Tf'</font> is <a href="funct_2.html#V2">onto</a> )
 
<b>by </b><i>, <a class="ref" href="fsm_1.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_1.html#T62">FUNCT_1:62</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:81_1_2"/>
<font color="Maroon">Tf'</font> is <a href="funct_2.html#V3">bijective</a>
 <b>by </b><i><a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>

<b>thus </b><a NAME="E10:81_1_2"/>
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">tfsm1</font> <a href="hidden.html#R1">=</a> <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">tfsm2</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T11" target="_self">Th11</a>, , , <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_2_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">tfsm2</font>;<br/><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font>;<br/><a NAME="E1:81_1_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">Tf'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/><b>thus </b>the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm1</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">s</font></span><a href="domain_1.html#K1">]</a></span> = 
<font color="Maroon">Tf'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">Tf</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> <span class="p3">(<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">s</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T56">FUNCT_1:56</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">tfsm2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">q</font>,<font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="fsm_1.html#T11" target="_self">Th11</a>, </i>
;<br/><b>thus </b><a NAME="E3:81_1_2_1"/>
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">tfsm1</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="Maroon">q</font></span>)</span>,<font color="Maroon">s</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> <font color="Maroon">q</font>,<font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T11" target="_self">Th11</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:81_1_2"/>
for <font color="Olive">q11</font> being  <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">tfsm2</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">tfsm2</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">tfsm1</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">tfsm2</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">tfsm1</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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