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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <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">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>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>











<b>assume </b><a NAME="E1:36"/><i><font color="Green">E24</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">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:36"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" target="_self">great_eq_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="measure2.html#K1">meet</a> <span class="p1">(<span class="default"><a href="measure3.html#K1">rng</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<a NAME="E1:36_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" target="_self">great_eq_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="measure2.html#K1">meet</a> <span class="p1">(<span class="default"><a href="measure3.html#K1">rng</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_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> ;<br/>

<b>assume </b><a NAME="E1:36_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" target="_self">great_eq_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:36_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K18" target="_self">great_eq_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:36_1_1"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a>  <a href="measure3.html#K1">rng</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:36_1_1_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="measure3.html#K1">rng</a> <font color="Maroon">F</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1_1_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="measure3.html#K1">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">m</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="E3:36_1_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:36_1_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="mesfunc1.html#E3:36_1_1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E5:36_1_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E8:36_1_1_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> &amp;  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:36_1_1_1_1"/>
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:36_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K2">"</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E2:36_1_1_1_1_1"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/>
<a NAME="E3:36_1_1_1_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E4:36_1_1_1_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1_1_1_1_1"/>
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:36_1_1_1_1"/><b>then </b>
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:36_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="mesfunc1.html#E1:36"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:36_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="txt" href="mesfunc1.html#E4:36_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:36_1_1_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="measure3.html#K1">rng</a> <font color="Maroon">F</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:36_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="measure2.html#K1">meet</a> <span class="p1">(<span class="default"><a href="measure3.html#K1">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:36_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" target="_self">great_eq_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="measure2.html#K1">meet</a> <span class="p1">(<span class="default"><a href="measure3.html#K1">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:36"/>
 <a href="measure2.html#K1">meet</a> <span class="p1">(<span class="default"><a href="measure3.html#K1">rng</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" target="_self">great_eq_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<a NAME="E1:36_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="measure2.html#K1">meet</a> <span class="p1">(<span class="default"><a href="measure3.html#K1">rng</a> <font color="Maroon">F</font></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" target="_self">great_eq_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:36_2_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="measure2.html#K1">meet</a> <span class="p1">(<span class="default"><a href="measure3.html#K1">rng</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<a NAME="E2:36_2_1"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> &amp;  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:36_2_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<a NAME="E2:36_2_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:36_2_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="measure3.html#K1">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:36_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E5:36_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:36_2_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:36_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> &amp;  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> ) )
 <b>by </b><i><a class="txt" href="mesfunc1.html#E1:36"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E4:36_2_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<a NAME="E5:36_2_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:36_2_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="measure3.html#K1">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:36_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 1
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E8:36_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:36_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E11:36_2_1"/>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:36_2_1_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon">y1</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E2:36_2_1_2_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> &amp;  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:36_2_1_2_1"/>
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_2_1_2_2_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:36_2_1_2_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_2_1_2_2_1_1"/>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:36_2_1_2_2_1_2"/>
not <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:36_2_1_2_2_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
not  <a href="supinf_1.html#K1">+infty</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/><a NAME="E3:36_2_1_2_2_1_2"/>
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="measure6.html#T15">MEASURE6:15</a></i>;<br/><a NAME="E5:36_2_1_2_2_1_2"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1_2_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:36_2_1_2_2_1_2_1"/>
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:36_2_1_2_2_1_2_1"/>
<font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E6:36_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:36_2_1_2_2_1_2"/>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:36_2_1_2"/>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:36_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K18" target="_self">great_eq_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E13:36_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" target="_self">great_eq_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:36_2"/>
 <a href="measure2.html#K1">meet</a> <span class="p1">(<span class="default"><a href="measure3.html#K1">rng</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" target="_self">great_eq_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:36"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18" target="_self">great_eq_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="measure2.html#K1">meet</a> <span class="p1">(<span class="default"><a href="measure3.html#K1">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
