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<b>let </b><font color="Maroon">I</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">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="fsm_1.html#L1">FSM</a> of <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E1:23"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">S</font> is <a href="fsm_2.html#V3" target="_self">regular</a> &amp; <font color="Maroon">S</font> is <a href="fsm_2.html#V1" target="_self">calculating_type</a> )
 ;<br/>

<b>let </b><font color="Maroon">s1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">s2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">S</font>;<br/>





<b>assume </b><a NAME="E2:23"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</font>
 ;<br/>

<b>set </b><font color="Maroon">w1</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s1</font></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">w2</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s2</font></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<a NAME="E3:23"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s1</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon">j</font> =  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s1</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<b>consider </b><font color="Maroon">WI</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>, <font color="Maroon">QI</font> being   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">S</font>, <font color="Maroon">QI1</font> being   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E6:23"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">WI</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s1</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">QI</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s1</font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">QI1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s1</font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">WI</font> <a href="fsm_1.html#K1">-succ_of</a> <font color="Maroon">QI</font> <a href="hidden.html#R1">=</a> <font color="Maroon">QI1</font> )
 <b>by </b><i>, <a class="ref" href="fsm_1.html#D2">FSM_1:def 2</a></i>;<br/>
<a NAME="E7:23"/>
<font color="Maroon">WI</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E8:23"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">QI1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="fsm_1.html#K1">-succ_of</a> <font color="Maroon">q</font>
 
<b>by </b><i>, , <a class="ref" href="fsm_1.html#D2">FSM_1:def 2</a></i>;<br/>
<a NAME="E9:23"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s2</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h</font> =  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s2</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<b>consider </b><font color="Maroon">WI2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>, <font color="Maroon">QI2</font> being   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">S</font>, <font color="Maroon">QI12</font> being   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E12:23"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">WI2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s2</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> &amp; <font color="Maroon">QI2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s2</font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> &amp; <font color="Maroon">QI12</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s2</font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">WI2</font> <a href="fsm_1.html#K1">-succ_of</a> <font color="Maroon">QI2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">QI12</font> )
 <b>by </b><i>, <a class="ref" href="fsm_1.html#D2">FSM_1:def 2</a></i>;<br/>
<a NAME="E13:23"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">QI2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 
<b>by </b><i>, , <a class="ref" href="fsm_1.html#D2">FSM_1:def 2</a></i>;<br/>
<a NAME="E14:23"/>
<font color="Maroon">WI2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E15:23"/><b>then </b>
<font color="Maroon">s1</font> <a href="fsm_1.html#K1">-succ_of</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">s2</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, , , , , , , , </i>;<br/>
<b>hence </b><a NAME="E16:23"/>
the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">s1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">s2</font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/>


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