<?xml version="1.0"?>
<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> ,  <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3">REAL</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:31_1_1"/><i><font color="Green">E27</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>   ex <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon">GG</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> <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E3:31_1_1"/><i><font color="Green">E28</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">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">GG</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="measure7.html#E1:31_1_1"><i><font color="Green">E27</font></i></a>);<br/></i>
<a NAME="E4:31_1_1"/>
<font color="Maroon">GG</font> is    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_1_1_1"/><i><font color="Green">E30</font></i>: 
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</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="relset_1.html#K5">rng</a> <font color="Maroon">GG</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:31_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 ;<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="E3:31_1_1_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.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">n1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:31_1_1_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">n1</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">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> )
 <b>by </b><i><a class="txt" href="measure7.html#E1:31_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E7:31_1_1_1_1"/>
<font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n1</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">AA</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:31_1_1_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font> &amp; <font color="Maroon">AA</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="measure7.html#E1:31_1_1_1"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">n2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:31_1_1_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n1</font></span>)</span> &amp; <font color="Maroon">AA</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> )
 <b>by </b><i><a class="txt" href="measure7.html#E3:31_1_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n2</font> = <font color="Maroon">n2</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="measure7.html#E5:31_1_1_1_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E13:31_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">n1</font>,<font color="Maroon">n2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="funcop_1.html#K9">rng</a> <font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:31_1_1_1_1"/><i><font color="Green">E42</font></i>: 
( <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; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">n1</font>,<font color="Maroon">n2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E17:31_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">n1</font>,<font color="Maroon">n2</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, <a class="txt" href="measure7.html#E1:31"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E18:31_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> &amp; <span class="p1">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E19:31_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GG</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">GG</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">GG</font> )
 
<b>by </b><i><a class="txt" href="measure7.html#E3:31_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="measure7.html#E3:31_1_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="measure7.html#E5:31_1_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<b>hence </b><a NAME="E20:31_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">GG</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:31_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">GG</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> is   <a href="measure5.html#NM1">Interval</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_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="E1:31_1_1_1_2"/>
<font color="Maroon">GG</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E3:31_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:31_1_1_1_2"/>
<font color="Maroon">GG</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> is   <a href="measure5.html#NM1">Interval</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:31_1_1_1"/>
<font color="Maroon">GG</font> is    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">GG</font> = <font color="Maroon">GG</font> as    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">GG</font>
;<br/>

<b>thus </b><a NAME="E6:31_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">GG</font> <a href="measure7.html#K3" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure7.html#E3:31_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>


</div>
