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

<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="E1:20"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">s1</font>, <font color="Olive">s2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><br/> for <font color="Olive">q</font> being  <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">S</font> holds  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="Olive">q</font>,<font color="Olive">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="Olive">q</font>,<font color="Olive">s2</font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/>

<a NAME="E2:20"/>
for <font color="Olive">j</font> being 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> <br/> for <font color="Olive">w1</font>, <font color="Olive">w2</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">I</font>  st <font color="Olive">w1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">w2</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">w1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">w2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Olive">w1</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Olive">w2</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be  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> ;<br/><b>let </b><font color="Maroon">w1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">w2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">I</font>;<br/>





<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">w1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<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> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">w1</font>, <font color="Olive">w2</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">I</font>  st <font color="Olive">w1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">w2</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">w1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">w2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Olive">w1</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Olive">w2</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:20_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">w2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">I</font>;<br/>



<a NAME="E1:20_1_1"/>
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w1</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D2">FSM_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E2:20_1_1"/>
( <font color="Maroon">w1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font> <a href="funct_1.html#K1">.</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<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> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 implies <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w1</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w2</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 )
 <b>by </b><i><a class="ref" href="fsm_1.html#D2">FSM_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:20_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">h</font> being 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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">h</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">h</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">h</font> be  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> ;<br/>

<b>assume </b><a NAME="E1:20_1_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">w1</font>, <font color="Olive">w2</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">I</font>  st <font color="Olive">w1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">w2</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">w1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">w2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Olive">w1</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Olive">w2</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font>
 ;<br/>

<b>let </b><font color="Maroon">w1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">w2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E2:20_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">w1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<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> 1 &amp; <font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<a NAME="E3:20_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w1</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E4:20_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<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> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:20_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>then 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="E7:20_1_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">WI</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</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> <font color="Maroon">w1</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</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> <font color="Maroon">w1</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</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">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="E8:20_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w2</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E9:20_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E10:20_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>then 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="E12:20_1_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">WI2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font> <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> <font color="Maroon">w2</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</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> <font color="Maroon">w2</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</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="E13:20_1_2"/>
<font color="Maroon">QI</font> <a href="hidden.html#R1">=</a> <font color="Maroon">QI2</font>
 
<b>by </b><i>, , , , , </i>;<br/>
<a NAME="E14:20_1_2"/><b>then </b>
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">QI</font>,<font color="Maroon">WI</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">QI2</font>,<font color="Maroon">WI2</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:20_1_2"/><b>then </b>
<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> 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">QI2</font>,<font color="Maroon">WI2</font></span><a href="domain_1.html#K1">]</a></span>
 
;<br/>
<b>hence </b><a NAME="E16:20_1_2"/>
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w1</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w2</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</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>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:20_1"/>
for <font color="Olive">j</font> being 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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>]
 
<b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E5:20_1"/>
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w1</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w2</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20"/>
<font color="Maroon">S</font> is <a href="fsm_2.html#V1" target="_self">calculating_type</a>
 <b>by </b><i/>;<br/>


</div>
