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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a>  <a href="fsm_1.html#L2" target="_self">Mealy-FSM</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:90_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="fsm_1.html#V5" target="_self">reduced</a>
 ;<br/><a NAME="E2:90_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>, <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#R7" target="_self">-are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="fsm_1.html#T64" target="_self">Th64</a></i>;<br/><b>hence </b><a NAME="E3:90_1"/>
ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a>  <a href="fsm_1.html#L2" target="_self">Mealy-FSM</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>3</sub></font>, <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Olive">b<sub>1</sub></font> <a href="fsm_1.html#R7" target="_self">-are_isomorphic</a> 
 ;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>5</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a>  <a href="fsm_1.html#L2" target="_self">Mealy-FSM</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><a NAME="E2:90"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>3</sub></font>, <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>5</sub></font> <a href="fsm_1.html#R7" target="_self">-are_isomorphic</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">c<sub>3</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="fsm_1.html#D20" target="_self">FSM_1:def 20</a><br/>

<b>assume </b><a NAME="E3:90"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:90"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> is <a href="funct_2.html#V3">bijective</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>9</sub></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">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> &amp; the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:90"><i><font color="Green">E67</font></i></a>, <a class="ref" href="fsm_1.html#D19" target="_self">Def19</a></i>;<br/>
<a NAME="E6:90"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E5:90"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E7:90"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E3:90"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/>
<a NAME="E8:90"/><b>then </b>
not <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 
<b>by </b><i><a class="ref" href="fsm_1.html#T62" target="_self">Th62</a></i>;<br/>
<b>hence </b><a NAME="E9:90"/>
not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 <b>by </b><i><a class="txt" href="fsm_1.html#E5:90"><i><font color="Green">E69</font></i></a>, <a class="ref" href="fsm_1.html#T61" target="_self">Th61</a></i>;<br/>


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