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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> ; <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="struct_0.html#V8" title="STRUCT_0:attr.8">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/> for <font color="Olive" title="b2">CRtfsm1</font>, <font color="Olive" title="b3">CRtfsm2</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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/> for <font color="Olive" title="b4">q1u</font>, <font color="Olive" title="b5">q2u</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Olive" title="b1">tfsm</font><br/> for <font color="Olive" title="b6">crq11</font>, <font color="Olive" title="b7">crq12</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Olive" title="b2">CRtfsm1</font>  st <font color="Olive" title="b6">crq11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q1u</font> &amp; <font color="Olive" title="b7">crq12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">q2u</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">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="Olive" title="b3">CRtfsm2</font> &amp; <font color="Olive" title="b2">CRtfsm1</font>,<font color="Olive" title="b3">CRtfsm2</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a>  &amp; <font color="Olive" title="b1">tfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">CRtfsm1</font> <a href="fsm_1.html#K13" title="FSM_1:func.13">-Mealy_union</a> <font color="Olive" title="b3">CRtfsm2</font> &amp; not <font color="Olive" title="b6">crq11</font>,<font color="Olive" title="b7">crq12</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a>  holds <br/> not <font color="Olive" title="b4">q1u</font>,<font color="Olive" title="b5">q2u</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> </span><br/><b>let </b><font color="Maroon" title="c3">tfsm</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">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>; <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">CRtfsm1</font>, <font color="Olive" title="b2">CRtfsm2</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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/> for <font color="Olive" title="b3">q1u</font>, <font color="Olive" title="b4">q2u</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c3">tfsm</font><br/> for <font color="Olive" title="b5">crq11</font>, <font color="Olive" title="b6">crq12</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Olive" title="b1">CRtfsm1</font>  st <font color="Olive" title="b5">crq11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">q1u</font> &amp; <font color="Olive" title="b6">crq12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q2u</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">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="Olive" title="b2">CRtfsm2</font> &amp; <font color="Olive" title="b1">CRtfsm1</font>,<font color="Olive" title="b2">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="Olive" title="b1">CRtfsm1</font> <a href="fsm_1.html#K13" title="FSM_1:func.13">-Mealy_union</a> <font color="Olive" title="b2">CRtfsm2</font> &amp; not <font color="Olive" title="b5">crq11</font>,<font color="Olive" title="b6">crq12</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a>  holds <br/> not <font color="Olive" title="b3">q1u</font>,<font color="Olive" title="b4">q2u</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> </span><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#V2" title="STRUCT_0:attr.2">empty</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>; <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">q1u</font>, <font color="Olive" title="b2">q2u</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c3">tfsm</font><br/> for <font color="Olive" title="b3">crq11</font>, <font color="Olive" title="b4">crq12</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c4">CRtfsm1</font>  st <font color="Olive" title="b3">crq11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q1u</font> &amp; <font color="Olive" title="b4">crq12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q2u</font> &amp; 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> &amp; not <font color="Olive" title="b3">crq11</font>,<font color="Olive" title="b4">crq12</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a>  holds <br/> not <font color="Olive" title="b1">q1u</font>,<font color="Olive" title="b2">q2u</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> </span><br/><b>let </b><font color="Maroon" title="c6">q1u</font>, <font color="Maroon" title="c7">q2u</font> be   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c3">tfsm</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">crq11</font>, <font color="Olive" title="b2">crq12</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c4">CRtfsm1</font>  st <font color="Olive" title="b1">crq11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q1u</font> &amp; <font color="Olive" title="b2">crq12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q2u</font> &amp; 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> &amp; not <font color="Olive" title="b1">crq11</font>,<font color="Olive" title="b2">crq12</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a>  holds <br/> not <font color="Maroon" title="c6">q1u</font>,<font color="Maroon" title="c7">q2u</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> </span><br/><b>let </b><font color="Maroon" title="c8">crq11</font>, <font color="Maroon" title="c9">crq12</font> be   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c4">CRtfsm1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">crq11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q1u</font> &amp; <font color="Maroon" title="c9">crq12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q2u</font> &amp; 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> &amp; not <font color="Maroon" title="c8">crq11</font>,<font color="Maroon" title="c9">crq12</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a>  implies  not <font color="Maroon" title="c6">q1u</font>,<font color="Maroon" title="c7">q2u</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a>  )</span><br/>















<b>set </b><font color="Maroon" title="c10">rtfsm1</font> = <font color="Maroon" title="c4">CRtfsm1</font>;<br/>
<b>set </b><font color="Maroon" title="c11">rtfsm2</font> = <font color="Maroon" title="c5">CRtfsm2</font>;<br/>
<b>set </b><font color="Maroon" title="c12">q1</font> = <font color="Maroon" title="c8">crq11</font>;<br/>
<b>set </b><font color="Maroon" title="c13">q2</font> = <font color="Maroon" title="c9">crq12</font>;<br/>
<b>assume </b><a NAME="E1:106"/><i><font color="Green" title="E78">A1</font></i>: 
( <font color="Maroon" title="c8">crq11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q1u</font> &amp; <font color="Maroon" title="c9">crq12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q2u</font> &amp; 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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">crq11</font>,<font color="Maroon" title="c9">crq12</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a>  or  not <font color="Maroon" title="c6">q1u</font>,<font color="Maroon" title="c7">q2u</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a>  )</span><br/>

<b>assume </b><a NAME="E2:106"/>
not <font color="Maroon" title="c8">crq11</font>,<font color="Maroon" title="c9">crq12</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"> not <font color="Maroon" title="c6">q1u</font>,<font color="Maroon" title="c7">q2u</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> </span><br/>

<b>then consider </b><font color="Maroon" title="c14">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">IAlph</font><b> such that </b><br/><a NAME="E4:106"/><i><font color="Green" title="E79">A2</font></i>: 
<font color="Maroon" title="c8">crq11</font>,<font color="Maroon" title="c14">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">crq12</font>,<font color="Maroon" title="c14">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 <b>by </b><i><a class="ref" href="fsm_1.html#D10" target="_self" title="FSM_1:def.10">Def10</a></i>;<br/>
<a NAME="E5:106"/>
<font color="Maroon" title="c6">q1u</font>,<font color="Maroon" title="c14">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="c8">crq11</font>,<font color="Maroon" title="c14">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 
<b>by </b><i><a class="txt" href="fsm_1.html#E1:106"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="fsm_1.html#T72" target="_self" title="FSM_1:th.72">Th72</a></i>;<br/>
<a NAME="E6:106"/><b>then </b>
<font color="Maroon" title="c6">q1u</font>,<font color="Maroon" title="c14">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">q2u</font>,<font color="Maroon" title="c14">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 
<b>by </b><i><a class="txt" href="fsm_1.html#E1:106"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="fsm_1.html#E4:106"><i><font color="Green" title="E79">A2</font></i></a>, <a class="ref" href="fsm_1.html#T72" target="_self" title="FSM_1:th.72">Th72</a></i>;<br/>
<b>hence </b><a NAME="E7:106"/>
not <font color="Maroon" title="c6">q1u</font>,<font color="Maroon" title="c7">q2u</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 <b>by </b><i><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/>


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