<?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="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>consider </b><font color="Maroon" title="c3">F</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><b> such that </b><br/><a NAME="E2:42"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c2">S</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="c3">F</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/>
<a NAME="E3:42"/><i><font color="Green" title="E24">A2</font></i>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">F</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:42"/><i><font color="Green" title="E25">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:42_1"/><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>

<b>take </b>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>
;<br/>

<b>thus </b><a NAME="E2:42_1"/>
( <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )
 <b>by </b><i><a class="txt" href="measure1.html#E2:42"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="measure1.html#E1:42_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">V</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  );<br/>
<a NAME="E5:42"/><i><font color="Green" title="E26">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:42_2"/>
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">V</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:42_2"/><i><font color="Green" title="E27">A6</font></i>: 
( <font color="Maroon" title="c5">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c5">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )
 <b>by </b><i><a class="txt" href="measure1.html#E4:42"><i><font color="Green" title="E25">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">V</font> = <font color="Maroon" title="c5">V</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="measure1.html#E3:42_2"><i><font color="Green" title="E27">A6</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
;<br/>

<a NAME="E5:42_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="measure1.html#E3:42_2"><i><font color="Green" title="E27">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:42_2"/>
( <font color="Maroon" title="c6">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c6">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> ] )
 <b>by </b><i><a class="txt" href="measure1.html#E3:42_2"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:42"/><i><font color="Green" title="E27">A7</font></i>: 
 ex <font color="Olive" title="b1">G</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> ,<font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c2">S</font> st <br/>for <font color="Olive" title="b2">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">n</font>,<font color="Olive" title="b1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="measure1.html#E5:42"><i><font color="Green" title="E26">A5</font></i></a>);<br/></i>
<a NAME="E7:42"/>
 ex <font color="Olive" title="b1">G</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 <font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c2">S</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">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c4">G</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> ,<font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E2:42_3"/><i><font color="Green" title="E28">A8</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b2">V</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 <b>by </b><i><a class="txt" href="measure1.html#E6:42"><i><font color="Green" title="E27">A7</font></i></a></i>;<br/>
<a NAME="E3:42_3"/><i><font color="Green" title="E29">A9</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:42_3"/><i><font color="Green" title="E30">A10</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:42_3_1"/><i><font color="Green" title="E31">A11</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c2">S</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c5">x</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:42_3_1"/><i><font color="Green" title="E32">A12</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="measure1.html#E1:42_3_1"><i><font color="Green" title="E31">A11</font></i></a>, <a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:42_3_1"/><i><font color="Green" title="E33">A13</font></i>: 
( <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 <b>by </b><i><a class="txt" href="measure1.html#E2:42"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="measure1.html#E3:42"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="measure1.html#E3:42_3_1"><i><font color="Green" title="E32">A12</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:42_3_1"/>
 ex <font color="Olive" title="b1">V</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<b>by </b><i><a class="txt" href="measure1.html#E2:42_3"><i><font color="Green" title="E28">A8</font></i></a>, <a class="txt" href="measure1.html#E5:42_3_1"><i><font color="Green" title="E33">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:42_3_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">G</font>
 <b>by </b><i><a class="txt" href="measure1.html#E3:42_3"><i><font color="Green" title="E29">A9</font></i></a>, <a class="txt" href="measure1.html#E5:42_3_1"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c5">G</font> = <font color="Maroon" title="c4">G</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#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">G</font>
;<br/>

<b>thus </b><a NAME="E6:42_3"/>
<font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c2">S</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="c5">G</font>
 <b>by </b><i><a class="txt" href="measure1.html#E4:42_3"><i><font color="Green" title="E30">A10</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:42"/>
<font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c2">S</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>
