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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">tfsm</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">IAlph</font>,<font color="Maroon">OAlph</font>;<br/><b>let </b><font color="Maroon">CRtfsm1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a> <a href="fsm_1.html#V5" target="_self">reduced</a> <a href="fsm_1.html#V7" target="_self">connected</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">CRtfsm2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a> <a href="fsm_1.html#V5" target="_self">reduced</a> <a href="fsm_1.html#V7" target="_self">connected</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>set </b><font color="Maroon">rtfsm1</font> = <font color="Maroon">CRtfsm1</font>;<br/>
<b>set </b><font color="Maroon">rtfsm2</font> = <font color="Maroon">CRtfsm2</font>;<br/>
<b>assume </b><a NAME="E1:112"/><i><font color="Green">E22</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> <a href="subset_1.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font> &amp; <font color="Maroon">CRtfsm1</font>,<font color="Maroon">CRtfsm2</font> <a href="fsm_1.html#R4" target="_self">-are_equivalent</a>  &amp; <font color="Maroon">tfsm</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font> )
 ;<br/>

<b>consider </b><font color="Maroon">tfsm2</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">IAlph</font>,<font color="Maroon">OAlph</font><b> such that </b><br/><a NAME="E3:112"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">CRtfsm2</font>, <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">tfsm2</font> <a href="fsm_1.html#R7" target="_self">-are_isomorphic</a> 
 <b>by </b><i/>;<br/>
<b>given </b><font color="Maroon">Q</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">tfsm</font></span>)</span>, <font color="Maroon">q1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Q</font>, <font color="Maroon">q2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Q</font><b> such that </b><a NAME="E4:112"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font> )
 ;<br/>

<b>set </b><font color="Maroon">tfsm2r</font> =  <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">tfsm2</font>;<br/>
<b>consider </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">CRtfsm2</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">tfsm2</font></span>)</span><b> such that </b><br/><a NAME="E6:112"/><i><font color="Green">E39</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">CRtfsm2</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">tfsm2</font></span>)</span> &amp; ( for <font color="Olive">q</font> being  <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">CRtfsm2</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">CRtfsm2</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 <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">tfsm2</font></span>)</span> <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">CRtfsm2</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 <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">tfsm2</font></span>)</span> <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> ) ) )
 <b>by </b><i>, <a class="ref" href="fsm_1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E7:112"/>
( <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="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E8:112"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">Tf</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <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 <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">tfsm2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E9:112"/><i><font color="Green">E42</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">CRtfsm2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:112"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">Tf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> 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">tfsm2</font></span>)</span> &amp; <font color="Maroon">Tf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> 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">tfsm2</font></span>)</span> &amp; <font color="Maroon">Tf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Tf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q2</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Tq1</font> = <font color="Maroon">Tf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q1</font>, <font color="Maroon">Tq2</font> = <font color="Maroon">Tf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q2</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">tfsm2</font></span>)</span> <b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:112"/><i><font color="Green">E44</font></i>: 
not <font color="Maroon">Tq1</font>,<font color="Maroon">Tq2</font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">rtfsm</font> =  <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">tfsm</font>;<br/>
<a NAME="E13:112"/><i><font color="Green">E45</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">q1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">tfsm</font> <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q2</font> = <font color="Maroon">q2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">tfsm</font> <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E16:112"/><i><font color="Green">E46</font></i>: 
 <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font> is <a href="fsm_1.html#V4" target="_self">final</a>
 
<b>by </b><i/>;<br/>
<a NAME="E17:112"/><i><font color="Green">E47</font></i>: 
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">tfsm</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">tfsm</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E19:112"/>
<font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E20:112"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 
<b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">q1'</font> = <font color="Maroon">q1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">CRtfsm2</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">q2'</font> = <font color="Maroon">q2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">CRtfsm2</font> <b>by </b><i/>;<br/>
<a NAME="E23:112"/>
not <font color="Maroon">q1'</font>,<font color="Maroon">q2'</font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 
<b>by </b><i>, , <a class="ref" href="fsm_1.html#T12" target="_self">Th12</a></i>;<br/>
<b>hence </b><a NAME="E24:112"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="fsm_1.html#E20:112"><i><font color="Green">E79</font></i></a></i>;<br/>


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