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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span>;<br/>
<a NAME="E1:31_1_1"/><i><font color="Green" title="E27">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c5">GG</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> <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> <b> such that </b><br/><a NAME="E3:31_1_1"/><i><font color="Green" title="E28">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c5">GG</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="measure7.html#E1:31_1_1"><i><font color="Green" title="E27">A2</font></i></a>);<br/></i>
<a NAME="E4:31_1_1"/>
<font color="Maroon" title="c5">GG</font> is    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">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" title="E29">A4</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">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" title="c6">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="E1:31_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">F</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_1_1_1_1"/><i><font color="Green" title="E30">A5</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c7">A</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="c2">F</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">n1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:31_1_1_1_1"/><i><font color="Green" title="E31">A6</font></i>: 
( <font color="Maroon" title="c8">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n1</font> )
 <b>by </b><i><a class="txt" href="measure7.html#E3:31_1_1_1_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">n1</font> = <font color="Maroon" title="c8">n1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="measure7.html#E5:31_1_1_1_1"><i><font color="Green" title="E31">A6</font></i></a></i>;<br/>
<a NAME="E7:31_1_1_1_1"/>
<font color="Maroon" title="c2">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">n1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c9">n1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="measure7.html#D2" target="_self" title="MEASURE7:def.2">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">AA</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:31_1_1_1_1"/><i><font color="Green" title="E32">A7</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">AA</font> &amp; <font color="Maroon" title="c10">AA</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="measure7.html#E3:31_1_1_1_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="measure7.html#E5:31_1_1_1_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">n2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:31_1_1_1_1"/><i><font color="Green" title="E33">A8</font></i>: 
( <font color="Maroon" title="c11">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> &amp; <font color="Maroon" title="c10">AA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n2</font> )
 <b>by </b><i><a class="txt" href="measure7.html#E9:31_1_1_1_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">n2</font> = <font color="Maroon" title="c11">n2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="measure7.html#E11:31_1_1_1_1"><i><font color="Green" title="E33">A8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E14:31_1_1_1_1"/><i><font color="Green" title="E34">A9</font></i>: 
( <font color="Maroon" title="c13">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">H</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">n1</font>,<font color="Maroon" title="c12">n2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> )
 <b>by </b><i><a class="txt" href="measure7.html#E1:31"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">n</font> = <font color="Maroon" title="c13">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="measure7.html#E14:31_1_1_1_1"><i><font color="Green" title="E34">A9</font></i></a></i>;<br/>
<a NAME="E16:31_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">n</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">n1</font>,<font color="Maroon" title="c12">n2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="measure7.html#E14:31_1_1_1_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="measure7.html#D12" target="_self" title="MEASURE7:def.12">Def12</a></i>;<br/>
<a NAME="E17:31_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n1</font> &amp; <span class="p1">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n2</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E18:31_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">GG</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c14">n</font> &amp; <font color="Maroon" title="c5">GG</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c14">n</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="c5">GG</font> )
 
<b>by </b><i><a class="txt" href="measure7.html#E3:31_1_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="measure7.html#E9:31_1_1_1_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="measure7.html#E11:31_1_1_1_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/>
<b>hence </b><a NAME="E19:31_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">GG</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:31_1_1_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c5">GG</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">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" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:31_1_1_1_2"/>
<font color="Maroon" title="c5">GG</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E3:31_1_1"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:31_1_1_1_2"/>
<font color="Maroon" title="c5">GG</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c6">n</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:31_1_1_1"/>
<font color="Maroon" title="c5">GG</font> is    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure7.html#E1:31_1_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="measure7.html#D2" target="_self" title="MEASURE7:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">GG</font> = <font color="Maroon" title="c5">GG</font> as    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">F</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c6">GG</font>
;<br/>

<b>thus </b><a NAME="E6:31_1_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c6">GG</font> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure7.html#E3:31_1_1"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/>


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