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<b>let </b><font color="Maroon" title="c1">I</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="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="fsm_1.html#L1" title="FSM_1:struct.1">FSM</a> of <font color="Maroon" title="c1">I</font>;<br/>



<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E13">A1</font></i>: 
( <font color="Maroon" title="c2">S</font> is <a href="fsm_2.html#V3" title="FSM_2:attr.3">regular</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="fsm_2.html#V1" title="FSM_2:attr.1">calculating_type</a> )
 ;<br/>

<b>let </b><font color="Maroon" title="c3">s1</font>, <font color="Maroon" title="c4">s2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>;<br/><b>let </b><font color="Maroon" title="c5">q</font> be   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c2">S</font>;<br/>





<b>assume </b><a NAME="E2:23"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">w1</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">s1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon" title="c7">w2</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">s2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<a NAME="E3:23"/><i><font color="Green" title="E15">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">s1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">j</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">s1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">WI</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c10">QI</font>, <font color="Maroon" title="c11">QI1</font> being   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E6:23"/><i><font color="Green" title="E16">A4</font></i>: 
( <font color="Maroon" title="c9">WI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">s1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c10">QI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">s1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c11">QI1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">s1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c9">WI</font> <a href="fsm_1.html#K1" title="FSM_1:func.1">-succ_of</a> <font color="Maroon" title="c10">QI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">QI1</font> )
 <b>by </b><i><a class="txt" href="fsm_2.html#E3:23"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></i>;<br/>
<a NAME="E7:23"/>
<font color="Maroon" title="c9">WI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font>
 
<b>by </b><i><a class="txt" href="fsm_2.html#E3:23"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="fsm_2.html#E6:23"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E8:23"/><b>then </b><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c11">QI1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="fsm_1.html#K1" title="FSM_1:func.1">-succ_of</a> <font color="Maroon" title="c5">q</font>
 
<b>by </b><i><a class="txt" href="fsm_2.html#E3:23"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="fsm_2.html#E6:23"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></i>;<br/>
<a NAME="E9:23"/><i><font color="Green" title="E18">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">s2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">h</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">s2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">WI2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c14">QI2</font>, <font color="Maroon" title="c15">QI12</font> being   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E12:23"/><i><font color="Green" title="E19">A7</font></i>: 
( <font color="Maroon" title="c13">WI2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">s2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">h</font> &amp; <font color="Maroon" title="c14">QI2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">s2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">h</font> &amp; <font color="Maroon" title="c15">QI12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">s2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c13">WI2</font> <a href="fsm_1.html#K1" title="FSM_1:func.1">-succ_of</a> <font color="Maroon" title="c14">QI2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">QI12</font> )
 <b>by </b><i><a class="txt" href="fsm_2.html#E9:23"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></i>;<br/>
<a NAME="E13:23"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c14">QI2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font>
 
<b>by </b><i><a class="txt" href="fsm_2.html#E9:23"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="fsm_2.html#E12:23"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></i>;<br/>
<a NAME="E14:23"/>
<font color="Maroon" title="c13">WI2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font>
 
<b>by </b><i><a class="txt" href="fsm_2.html#E9:23"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="fsm_2.html#E12:23"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E15:23"/>
the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">s1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c4">s2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="fsm_2.html#E1:23"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="fsm_2.html#E2:23"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="fsm_2.html#E3:23"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="fsm_2.html#E6:23"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="fsm_2.html#E8:23"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="fsm_2.html#E9:23"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="fsm_2.html#E12:23"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="fsm_2.html#E13:23"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="fsm_2.html#T14" target="_self" title="FSM_2:th.14">Th14</a></i>;<br/>


</div>
