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<b>let </b><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> ;<br/><b>let </b><font color="Maroon" title="c3">tfsm</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</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>;<br/><b>let </b><font color="Maroon" title="c4">CRtfsm1</font>, <font color="Maroon" title="c5">CRtfsm2</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</a> <a href="fsm_1.html#V5" title="FSM_1:attr.5">reduced</a> <a href="fsm_1.html#V7" title="FSM_1:attr.7">connected</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>;<br/>









<b>set </b><font color="Maroon" title="c6">rtfsm1</font> = <font color="Maroon" title="c4">CRtfsm1</font>;<br/>
<b>set </b><font color="Maroon" title="c7">rtfsm2</font> = <font color="Maroon" title="c5">CRtfsm2</font>;<br/>
<b>assume </b><a NAME="E1:111"/><i><font color="Green" title="E81">A1</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> &amp; <font color="Maroon" title="c4">CRtfsm1</font>,<font color="Maroon" title="c5">CRtfsm2</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a>  &amp; <font color="Maroon" title="c3">tfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">CRtfsm1</font> <a href="fsm_1.html#K13" title="FSM_1:func.13">-Mealy_union</a> <font color="Maroon" title="c5">CRtfsm2</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">tfsm1</font> being  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</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><b> such that </b><br/><a NAME="E3:111"/><i><font color="Green" title="E82">A2</font></i>: 
<font color="Maroon" title="c4">CRtfsm1</font>, <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c8">tfsm1</font> <a href="fsm_1.html#R7" title="FSM_1:pred.7">-are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="fsm_1.html#T65" target="_self" title="FSM_1:th.65">Th65</a></i>;<br/>
<b>given </b><font color="Maroon" title="c9">Q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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">tfsm</font></span>)</span>, <font color="Maroon" title="c10">q1</font>, <font color="Maroon" title="c11">q2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">Q</font><b> such that </b><a NAME="E4:111"/><i><font color="Green" title="E83">A3</font></i>: 
( <font color="Maroon" title="c10">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="c4">CRtfsm1</font> &amp; <font color="Maroon" title="c11">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="c4">CRtfsm1</font> &amp; <font color="Maroon" title="c10">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">q2</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c12">tfsm1r</font> =  <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c8">tfsm1</font>;<br/>
<b>consider </b><font color="Maroon" title="c13">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="c4">CRtfsm1</font>,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="c8">tfsm1</font></span>)</span><b> such that </b><br/><a NAME="E6:111"/><i><font color="Green" title="E84">A4</font></i>: 
( <font color="Maroon" title="c13">Tf</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> &amp; <font color="Maroon" title="c13">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="c4">CRtfsm1</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="c8">tfsm1</font></span>)</span> &amp; ( for <font color="Olive" title="b1">q</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c4">CRtfsm1</font><br/> for <font color="Olive" title="b2">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="c13">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="c4">CRtfsm1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">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="c8">tfsm1</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">q</font></span>)</span>,<font color="Olive" title="b2">s</font> &amp; the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">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="c8">tfsm1</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">q</font></span>)</span>,<font color="Olive" title="b2">s</font> ) ) )
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:111"><i><font color="Green" title="E82">A2</font></i></a>, <a class="ref" href="fsm_1.html#D19" target="_self" title="FSM_1:def.19">Def19</a></i>;<br/>
<a NAME="E7:111"/>
( <font color="Maroon" title="c13">Tf</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c13">Tf</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 
<b>by </b><i><a class="txt" href="fsm_1.html#E6:111"><i><font color="Green" title="E84">A4</font></i></a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E8:111"/><b>then </b><i><font color="Green" title="E85">A5</font></i>: 
( <font color="Maroon" title="c13">Tf</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">Tf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="c8">tfsm1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E9:111"/><i><font color="Green" title="E86">A6</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c13">Tf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:111"/><b>then </b><i><font color="Green" title="E87">A7</font></i>: 
( <font color="Maroon" title="c13">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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 <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="c8">tfsm1</font></span>)</span> &amp; <font color="Maroon" title="c13">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">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 <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="c8">tfsm1</font></span>)</span> &amp; <font color="Maroon" title="c13">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">q2</font> )
 
<b>by </b><i><a class="txt" href="fsm_1.html#E4:111"><i><font color="Green" title="E83">A3</font></i></a>, <a class="txt" href="fsm_1.html#E8:111"><i><font color="Green" title="E85">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">Tq1</font> = <font color="Maroon" title="c13">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">q1</font>, <font color="Maroon" title="c15">Tq2</font> = <font color="Maroon" title="c13">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">q2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c8">tfsm1</font></span>)</span> <b>by </b><i><a class="txt" href="fsm_1.html#E4:111"><i><font color="Green" title="E83">A3</font></i></a>, <a class="txt" href="fsm_1.html#E8:111"><i><font color="Green" title="E85">A5</font></i></a>, <a class="txt" href="fsm_1.html#E9:111"><i><font color="Green" title="E86">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:111"/><i><font color="Green" title="E88">A8</font></i>: 
not <font color="Maroon" title="c14">Tq1</font>,<font color="Maroon" title="c15">Tq2</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 
<b>by </b><i><a class="txt" href="fsm_1.html#E10:111"><i><font color="Green" title="E87">A7</font></i></a>, <a class="ref" href="fsm_1.html#T62" target="_self" title="FSM_1:th.62">Th62</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">rtfsm</font> =  <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font>;<br/>
<a NAME="E13:111"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E1:111"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="fsm_1.html#D24" target="_self" title="FSM_1:def.24">Def24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">q1</font> = <font color="Maroon" title="c10">q1</font>, <font color="Maroon" title="c18">q2</font> = <font color="Maroon" title="c11">q2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">tfsm</font> <b>by </b><i><a class="txt" href="fsm_1.html#E4:111"><i><font color="Green" title="E83">A3</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E15:111"/><i><font color="Green" title="E89">A9</font></i>: 
 <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">tfsm</font> is <a href="fsm_1.html#V4" title="FSM_1:attr.4">final</a>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D15" target="_self" title="FSM_1:def.15">Def15</a></i>;<br/>
<a NAME="E16:111"/><i><font color="Green" title="E90">A10</font></i>: 
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">tfsm</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">tfsm</font>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D18" target="_self" title="FSM_1:def.18">Def18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">Q</font> = <font color="Maroon" title="c9">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">tfsm</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E18:111"/>
<font color="Maroon" title="c19">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="fsm_1.html#E16:111"><i><font color="Green" title="E90">A10</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E19:111"/><b>then </b><i><font color="Green" title="E91">A11</font></i>: 
<font color="Maroon" title="c17">q1</font>,<font color="Maroon" title="c18">q2</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 
<b>by </b><i><a class="txt" href="fsm_1.html#E15:111"><i><font color="Green" title="E89">A9</font></i></a>, <a class="txt" href="fsm_1.html#E16:111"><i><font color="Green" title="E90">A10</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self" title="FSM_1:def.14">Def14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">q1'</font> = <font color="Maroon" title="c17">q1</font>, <font color="Maroon" title="c21">q2'</font> = <font color="Maroon" title="c18">q2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">CRtfsm1</font> <b>by </b><i><a class="txt" href="fsm_1.html#E4:111"><i><font color="Green" title="E83">A3</font></i></a></i>;<br/>
<a NAME="E21:111"/>
not <font color="Maroon" title="c20">q1'</font>,<font color="Maroon" title="c21">q2'</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 
<b>by </b><i><a class="txt" href="fsm_1.html#E6:111"><i><font color="Green" title="E84">A4</font></i></a>, <a class="txt" href="fsm_1.html#E12:111"><i><font color="Green" title="E88">A8</font></i></a>, <a class="ref" href="fsm_1.html#T61" target="_self" title="FSM_1:th.61">Th61</a></i>;<br/>
<b>hence </b><a NAME="E22:111"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:111"><i><font color="Green" title="E81">A1</font></i></a>, <a class="txt" href="fsm_1.html#E19:111"><i><font color="Green" title="E91">A11</font></i></a>, <a class="ref" href="fsm_1.html#T76" target="_self" title="FSM_1:th.76">Th76</a></i>;<br/>


</div>
