<?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">M</font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">T</font> be    <a href="measure2.html#M1" target="_self">N_Measure_fam</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">F</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>assume </b><a NAME="E1:13"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="measure1.html#K5">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>consider </b><font color="Maroon">G</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="E3:13"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F</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="Maroon">G</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">F</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#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">H</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="E5:13"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F</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="Maroon">H</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">F</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">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> ) )
 <b>by </b><i/>;<br/>
<a NAME="E6:13"/><i><font color="Green">E40</font></i>: 
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">H</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> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_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="E1:13_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or  ex <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<a NAME="E2:13_1"/>
(  ex <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1 implies <font color="Maroon">H</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> <font color="Maroon">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:13_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:13_1_1"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:13_1_1"/>
<font color="Maroon">H</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> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:13_1"/>
<font color="Maroon">H</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> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:13"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_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="E1:13_2"/>
(  <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> &amp;  <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:13_2"/><b>then </b><i><font color="Green">E44</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>, <font color="Maroon">B</font> = <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E4:13_2"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="measure2.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E5:13_2"/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:13"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i/>;<br/>
<a NAME="E9:13"/>
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">m</font> holds <br/><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">m</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:13_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_3_1"/>
( ( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font> ) or  not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font> or  not <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_3_1_1"><b>suppose </b></a><a NAME="E1:13_3_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">n'</font> = <font color="Maroon">n</font>, <font color="Maroon">m'</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:13_3_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">n'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m'</font> implies <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E4:13_3_1_1"/>
( <font color="Maroon">m'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n'</font> implies <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E5:13_3_1_1"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="measure2.html#D1" target="_self">Def1</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_3_1_2"><b>suppose </b></a><a NAME="E1:13_3_1_2"/>
(  not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font> or  not <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font> )
 ;<br/><div class="add"><a NAME="E2:13_3_1_2"/><b>then </b>
( <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:13_3_1_2"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E10:13"/><b>then </b>
<font color="Maroon">H</font> is   <a href="measure1.html#NM4">Sep_Sequence</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="prob_2.html#D3">PROB_2:def 3</a></i>;<br/>
<a NAME="E11:13"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="measure1.html#K5">rng</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/>
<a NAME="E12:13"/>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K5">rng</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:13_4"/>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K5">rng</a> <font color="Maroon">H</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:13_4_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K5">rng</a> <font color="Maroon">H</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">E</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_4_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">E</font> <a href="hidden.html#R2">in</a>  <a href="measure1.html#K5">rng</a> <font color="Maroon">H</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">s</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:13_4_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font> &amp; <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="measure2.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="measure2.html#E1:13"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E7:13_4_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="measure2.html#T1" target="_self">Th1</a>, <a class="txt" href="measure2.html#E1:13"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E8:13_4_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="measure1.html#K5">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<b>hence </b><a NAME="E9:13_4_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="measure2.html#D1" target="_self">Def1</a>, <a class="txt" href="measure2.html#E3:13"><i><font color="Green">E21</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>

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

<b>assume </b><a NAME="E2:13_4"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">E</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:13_4"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">E</font> <a href="hidden.html#R2">in</a>  <a href="measure1.html#K5">rng</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">s</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:13_4"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="measure2.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="measure2.html#E1:13"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E8:13_4"/>
 ex <font color="Olive">s1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4_2"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:13_4_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="measure2.html#D1" target="_self">Def1</a>, <a class="txt" href="measure2.html#E1:13"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E2:13_4_2"/><b>then </b><i><font color="Green">E66</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
;<br/>
<a NAME="E3:13_4_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 
<b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="measure2.html#E3:13"><i><font color="Green">E21</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:13_4_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</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 <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 ;<br/>
<a NAME="E6:13_4_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 implies  ex <font color="Olive">s1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s1</font> )
 
<b>by </b><i>, <a class="txt" href="measure2.html#E5:13"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E7:13_4_2"/>
(  ex <font color="Olive">l</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">l</font> <a href="xcmplx_0.html#K2">+</a> 1 implies  ex <font color="Olive">s1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4_2_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">l</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:13_4_2_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:13_4_2_1"/><i><font color="Green">E71</font></i>: 
not <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_4_2_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">l</font>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:13_4_2_1_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E4:13_4_2_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="measure2.html#E5:13"><i><font color="Green">E22</font></i></a>, <a class="ref" href="measure2.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E5:13_4_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure2.html#T3" target="_self">Th3</a>, <a class="ref" href="measure2.html#T4" target="_self">Th4</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:13_4_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</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">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">l</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">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">l</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

<b>thus </b><a NAME="E5:13_4_2_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="measure2.html#E5:13"><i><font color="Green">E22</font></i></a>, <a class="ref" href="measure2.html#T3" target="_self">Th3</a>, , <a class="ref" href="measure2.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:13_4_2"/>
 ex <font color="Olive">s1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s1</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">s1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:13_4"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s1</font>
 ;<br/>
<a NAME="E11:13_4"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="measure1.html#K5">rng</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<b>hence </b><a NAME="E12:13_4"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K5">rng</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure2.html#E3:13"><i><font color="Green">E21</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:13"/>
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" target="_self">union</a> <font color="Maroon">T</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="supinf_2.html#T62">SUPINF_2:62</a></i>;<br/>


</div>
