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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">S</font>;<br/>





<b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><br/> for <font color="Olive">c</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Olive">c</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span>;<br/>
<a NAME="E2:5"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">c</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">A</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>  ex <font color="Olive">B</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>,<font color="Olive">A</font>,<font color="Olive">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/>



<b>reconsider </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> ;<br/>
<b>take </b>
<font color="Maroon">B</font>
;<br/>

<b>thus </b><a NAME="E2:5_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c</font>,<font color="Maroon">A</font>,<font color="Maroon">B</font>]
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">N</font> <a href="funct_2.html#K3">.</a> 0 as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> ;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E5:5"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">n</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">n</font>,<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>,<font color="Maroon">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>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>();<br/></i>
<a NAME="E6:5"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">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> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:5_2"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><br/> for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Olive">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Olive">s</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:5_2"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:5"/>
 ex <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">S</font> st <br/>( <font color="Olive">F</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_2.html#K3">.</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">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> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> ) )
 <b>by </b><i/>;<br/>


</div>
