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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">w1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">IAlph</font>;<br/><b>let </b><font color="Maroon">w2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">IAlph</font>;<br/><b>let </b><font color="Maroon">tfsm1</font> be  non <a href="struct_0.html#V3">empty</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">tfsm2</font> be  non <a href="struct_0.html#V3">empty</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">q11</font> be   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">tfsm1</font>;<br/><b>let </b><font color="Maroon">q12</font> be   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">tfsm1</font>;<br/><b>let </b><font color="Maroon">q21</font> be   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">tfsm2</font>;<br/><b>let </b><font color="Maroon">q22</font> be   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">tfsm2</font>;<br/>



















<b>assume </b><a NAME="E1:40"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">q11</font>,<font color="Maroon">w1</font> <a href="fsm_1.html#R1" target="_self">-leads_to</a> <font color="Maroon">q12</font> &amp; <font color="Maroon">q21</font>,<font color="Maroon">w1</font> <a href="fsm_1.html#R1" target="_self">-leads_to</a> <font color="Maroon">q22</font> &amp; <font color="Maroon">q12</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K4" target="_self">-response</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q22</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K4" target="_self">-response</a>  )
 ;<br/>

<b>assume </b><a NAME="E2:40"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">q11</font>,<span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font></span>)</span> <a href="fsm_1.html#K4" target="_self">-response</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q21</font>,<span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font></span>)</span> <a href="fsm_1.html#K4" target="_self">-response</a> 
 ;<br/>

<b>set </b><font color="Maroon">r12</font> = <font color="Maroon">q12</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K4" target="_self">-response</a> ;<br/>
<b>set </b><font color="Maroon">r22</font> = <font color="Maroon">q22</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K4" target="_self">-response</a> ;<br/>
<b>set </b><font color="Maroon">r1w1</font> = <font color="Maroon">q11</font>,<font color="Maroon">w1</font> <a href="fsm_1.html#K4" target="_self">-response</a> ;<br/>
<b>set </b><font color="Maroon">r2w1</font> = <font color="Maroon">q21</font>,<font color="Maroon">w1</font> <a href="fsm_1.html#K4" target="_self">-response</a> ;<br/>
<b>set </b><font color="Maroon">w</font> = <font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font>;<br/>
<b>set </b><font color="Maroon">r11</font> = <font color="Maroon">q11</font>,<span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font></span>)</span> <a href="fsm_1.html#K4" target="_self">-response</a> ;<br/>
<b>set </b><font color="Maroon">r21</font> = <font color="Maroon">q21</font>,<span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font></span>)</span> <a href="fsm_1.html#K4" target="_self">-response</a> ;<br/>
<a NAME="E3:40"/><i><font color="Green">E38</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">q21</font>,<font color="Maroon">w1</font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w1</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">q11</font>,<font color="Maroon">w1</font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w1</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:40"/><i><font color="Green">E39</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">q12</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w2</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">q22</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w2</font> )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:40"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">q12</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">q22</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>
<a NAME="E7:40"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q12</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q22</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> )
 
<b>by </b><i>, <a class="ref" href="fsm_1.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:40"/>
( <font color="Maroon">q11</font>,<span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font></span>)</span> <a href="fsm_1.html#K4" target="_self">-response</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q11</font>,<font color="Maroon">w1</font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">q12</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> &amp; <font color="Maroon">q21</font>,<span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font></span>)</span> <a href="fsm_1.html#K4" target="_self">-response</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q21</font>,<font color="Maroon">w1</font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">q22</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:40"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">q11</font>,<span class="p2">(<span class="default"><font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font></span>)</span> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q12</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">q21</font>,<span class="p2">(<span class="default"><font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font></span>)</span> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q22</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E10:40"/>
contradiction
 <b>by </b><i>, <a class="ref" href="fsm_1.html#T8" target="_self">Th8</a></i>;<br/>


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