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

<b>set </b><font color="Maroon">D</font> =  <a href="card_3.html#K4">product</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> =  <a href="circuit2.html#K6">Following</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E1:40_1_1"/>
(  ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="card_3.html#K4">product</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>,<font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>) ) ) &amp; ( for <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>  st  ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="card_3.html#K4">product</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>,<font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>) ) ) &amp;  ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="card_3.html#K4">product</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">y2</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>,<font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>) ) ) holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font> ) )
 
<b>from </b><i><a class="ref" href="recdef_1.html#S21">RECDEF_1:sch 21</a>();<br/></i>
<b>hence </b><a NAME="E2:40_1_1"/>
(  ex <font color="Olive">b<sub>1</sub></font> being  <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A</font> ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="card_3.html#K4">product</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A</font>  st  ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="card_3.html#K4">product</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> ) ) &amp;  ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="card_3.html#K4">product</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> ) ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/>


</div>
