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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">S</font>;<br/>





<a NAME="E1:51"/>
 ex <font color="Olive" title="b1">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> st  <a href="measure1.html#K5" title="MEASURE1:func.5">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<a NAME="E1:51_1"/>
 <a href="measure1.html#K5" title="MEASURE1:func.5">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">F</font> = <font color="Maroon" title="c3">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">F</font>
;<br/>

<b>thus </b><a NAME="E3:51_1"/>
 <a href="measure1.html#K5" title="MEASURE1:func.5">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">F</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:51"/>
 <a href="measure1.html#K5" title="MEASURE1:func.5">rng</a> <font color="Maroon" title="c3">F</font> is   <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#D14" title="SUPINF_2:def.14">SUPINF_2:def 14</a></i>;<br/>


</div>
