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

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>1</sub></font> );<br/>
<a NAME="E1:98_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="partfun1.html#T120">PARTFUN1:120</a></i>;<br/>
<a NAME="E2:98_1_1_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E4:98_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<b>hence </b><a NAME="E5:98_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:98_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>(<a class="txt" href="funct_7.html#E1:98_1_1"><i><font color="Green">E45</font></i></a>);<br/></i>
<a NAME="E5:98_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> is   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="partfun1.html#T120">PARTFUN1:120</a></i>;<br/>
<b>hence </b><a NAME="E6:98_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ex <font color="Olive">b<sub>4</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>1</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="funct_7.html#E4:98_1_1"><i><font color="Green">E46</font></i></a></i>;<br/>


</div>
