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<b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">IAlph</font>, <font color="Olive" title="b2">OAlph</font> being 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/> for <font color="Olive" title="b3">tfsm</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="fsm_1.html#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Olive" title="b1">IAlph</font>,<font color="Olive" title="b2">OAlph</font><br/> for <font color="Olive" title="b4">qa</font>, <font color="Olive" title="b5">qb</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Olive" title="b3">tfsm</font>  st <font color="Olive" title="b4">qa</font>,<font color="Olive" title="b5">qb</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a>  holds <br/><font color="Maroon" title="c1">k</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Olive" title="b4">qa</font>,<font color="Olive" title="b5">qb</font></span><br/><b>let </b><font color="Maroon" title="c2">IAlph</font>, <font color="Maroon" title="c3">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">tfsm</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="fsm_1.html#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Maroon" title="c2">IAlph</font>,<font color="Maroon" title="c3">OAlph</font><br/> for <font color="Olive" title="b2">qa</font>, <font color="Olive" title="b3">qb</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Olive" title="b1">tfsm</font>  st <font color="Olive" title="b2">qa</font>,<font color="Olive" title="b3">qb</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a>  holds <br/><font color="Maroon" title="c1">k</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Olive" title="b2">qa</font>,<font color="Olive" title="b3">qb</font></span><br/><b>let </b><font color="Maroon" title="c4">tfsm</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="fsm_1.html#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Maroon" title="c2">IAlph</font>,<font color="Maroon" title="c3">OAlph</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">qa</font>, <font color="Olive" title="b2">qb</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c4">tfsm</font>  st <font color="Olive" title="b1">qa</font>,<font color="Olive" title="b2">qb</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a>  holds <br/><font color="Maroon" title="c1">k</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Olive" title="b1">qa</font>,<font color="Olive" title="b2">qb</font></span><br/><b>let </b><font color="Maroon" title="c5">qa</font>, <font color="Maroon" title="c6">qb</font> be   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c4">tfsm</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">qa</font>,<font color="Maroon" title="c6">qb</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a>  implies <font color="Maroon" title="c1">k</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c5">qa</font>,<font color="Maroon" title="c6">qb</font> )</span><br/>











<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c5">qa</font>,<font color="Maroon" title="c6">qb</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">k</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c5">qa</font>,<font color="Maroon" title="c6">qb</font></span><br/>

<b>thus </b><a NAME="E2:53"/>
<font color="Maroon" title="c1">k</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c5">qa</font>,<font color="Maroon" title="c6">qb</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">w</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">IAlph</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="fsm_1.html#D11" target="_self" title="FSM_1:def.11">FSM_1:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> implies <font color="Maroon" title="c5">qa</font>,<font color="Maroon" title="c7">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">qb</font>,<font color="Maroon" title="c7">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  )</span><br/>

<b>assume </b><a NAME="E1:53_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">qa</font>,<font color="Maroon" title="c7">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">qb</font>,<font color="Maroon" title="c7">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span><br/>

<b>thus </b><a NAME="E2:53_1"/>
<font color="Maroon" title="c5">qa</font>,<font color="Maroon" title="c7">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">qb</font>,<font color="Maroon" title="c7">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="fsm_1.html#D10" target="_self" title="FSM_1:def.10">Def10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


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