<?xml version="1.0"?>
<div class="add">

<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">w1</font>, <font color="Maroon" title="c4">w2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">IAlph</font>;<br/><b>let </b><font color="Maroon" title="c5">tfsm1</font>, <font color="Maroon" title="c6">tfsm2</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</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="c7">q11</font>, <font color="Maroon" title="c8">q12</font> be   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c5">tfsm1</font>;<br/><b>let </b><font color="Maroon" title="c9">q21</font>, <font color="Maroon" title="c10">q22</font> be   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c6">tfsm2</font>;<br/>



















<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c7">q11</font>,<font color="Maroon" title="c3">w1</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> <font color="Maroon" title="c8">q12</font> &amp; <font color="Maroon" title="c9">q21</font>,<font color="Maroon" title="c3">w1</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> <font color="Maroon" title="c10">q22</font> &amp; <font color="Maroon" title="c8">q12</font>,<font color="Maroon" title="c4">w2</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="c10">q22</font>,<font color="Maroon" title="c4">w2</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  )
 ;<br/>

<b>assume </b><a NAME="E2:40"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c7">q11</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">w1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">w2</font></span>)</span> <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="c9">q21</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">w1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">w2</font></span>)</span> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c11">r12</font> = <font color="Maroon" title="c8">q12</font>,<font color="Maroon" title="c4">w2</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> ;<br/>
<b>set </b><font color="Maroon" title="c12">r22</font> = <font color="Maroon" title="c10">q22</font>,<font color="Maroon" title="c4">w2</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> ;<br/>
<b>set </b><font color="Maroon" title="c13">r1w1</font> = <font color="Maroon" title="c7">q11</font>,<font color="Maroon" title="c3">w1</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> ;<br/>
<b>set </b><font color="Maroon" title="c14">r2w1</font> = <font color="Maroon" title="c9">q21</font>,<font color="Maroon" title="c3">w1</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> ;<br/>
<b>set </b><font color="Maroon" title="c15">w</font> = <font color="Maroon" title="c3">w1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">w2</font>;<br/>
<b>set </b><font color="Maroon" title="c16">r11</font> = <font color="Maroon" title="c7">q11</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">w1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">w2</font></span>)</span> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> ;<br/>
<b>set </b><font color="Maroon" title="c17">r21</font> = <font color="Maroon" title="c9">q21</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">w1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">w2</font></span>)</span> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> ;<br/>
<a NAME="E3:40"/><i><font color="Green" title="E29">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">q21</font>,<font color="Maroon" title="c3">w1</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q11</font>,<font color="Maroon" title="c3">w1</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font> )
 
<b>by </b><i><a class="ref" href="fsm_1.html#D6" target="_self" title="FSM_1:def.6">Def6</a></i>;<br/>
<a NAME="E4:40"/><i><font color="Green" title="E30">A4</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q12</font>,<font color="Maroon" title="c4">w2</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">q22</font>,<font color="Maroon" title="c4">w2</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font> )
 
<b>by </b><i><a class="ref" href="fsm_1.html#D6" target="_self" title="FSM_1:def.6">Def6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">j</font> being    <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> <b> such that </b><br/><a NAME="E6:40"/><i><font color="Green" title="E31">A5</font></i>: 
( <font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">q12</font>,<font color="Maroon" title="c4">w2</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">q22</font>,<font color="Maroon" title="c4">w2</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>;<br/>
<a NAME="E7:40"/><i><font color="Green" title="E32">A6</font></i>: 
( <font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q12</font>,<font color="Maroon" title="c4">w2</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> &amp; <font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">q22</font>,<font color="Maroon" title="c4">w2</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="fsm_1.html#E4:40"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="fsm_1.html#E6:40"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:40"/>
( <font color="Maroon" title="c7">q11</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">w1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">w2</font></span>)</span> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q11</font>,<font color="Maroon" title="c3">w1</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q12</font>,<font color="Maroon" title="c4">w2</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> &amp; <font color="Maroon" title="c9">q21</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">w1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">w2</font></span>)</span> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">q21</font>,<font color="Maroon" title="c3">w1</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">q22</font>,<font color="Maroon" title="c4">w2</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="fsm_1.html#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="fsm_1.html#T26" target="_self" title="FSM_1:th.26">Th26</a></i>;<br/>
<a NAME="E9:40"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">q11</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">w1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">w2</font></span>)</span> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q12</font>,<font color="Maroon" title="c4">w2</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">q21</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">w1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">w2</font></span>)</span> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">q22</font>,<font color="Maroon" title="c4">w2</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> )
 
<b>by </b><i><a class="txt" href="fsm_1.html#E3:40"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="fsm_1.html#E7:40"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E10:40"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="fsm_1.html#E6:40"><i><font color="Green" title="E31">A5</font></i></a></i>;<br/>


</div>
