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

<a NAME="E1:20_1_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T10">PROB_1:10</a></i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">B</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">C</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:20_1_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</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> , <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E8:20_1_1"/><i><font color="Green">E22</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">B</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>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">n</font> holds <br/><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> ) )
 <b>by </b><i><a class="ref" href="measure1.html#T40">MEASURE1:40</a></i>;<br/>
<a NAME="E9:20_1_1"/><i><font color="Green">E40</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">F</font> <a href="prob_1.html#K1">.</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>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">n</font> holds <br/><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> ) )
 
<b>by </b><i>, , <a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/>
<a NAME="E10:20_1_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">S</font> <b>by </b><i><a class="ref" href="funct_2.html#T8">FUNCT_2:8</a></i>;<br/>
<a NAME="E12:20_1_1"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> is   <a href="measure1.html#NM3">N_Sub_set_fam</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="measure2.html#D1" target="_self">Def1</a>, <a class="ref" href="supinf_2.html#D14">SUPINF_2:def 14</a></i>;<br/>
<a NAME="E13:20_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">T</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> as    <a href="measure2.html#M1" target="_self">N_Measure_fam</a> of <font color="Maroon">S</font> <b>by </b><i><a class="txt" href="measure2.html#E3:20_1_1"><i><font color="Green">E20</font></i></a>, </i>;<br/>
<b>take </b>
<font color="Maroon">T</font>
;<br/>

<b>take </b>
<font color="Maroon">F</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="measure2.html#D2" target="_self">MEASURE2:def 2</a><br/>

<a NAME="E15:20_1_1"/>
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> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><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="E1:20_1_1_1"/>
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> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1"><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="E1:20_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<b>hence </b><a NAME="E2:20_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:20_1_1_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <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> <font color="Maroon">A</font> )
 
;<br/>
<b>hence </b><a NAME="E3:20_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</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>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:20_1_1"/>
( <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="measure1.html#K5">rng</a> <font color="Maroon">F</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">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <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>by </b><i>, , <a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/>


</div>
