<?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">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">A</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font>;<br/>









<b>assume </b><a NAME="E1:8"/><i><font color="Green">E19</font></i>: 
(  <a href="measure2.html#K1">meet</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</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>  holds  <font color="Maroon">A</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> ) )
 ;<br/>

<a NAME="E2:8"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K1">meet</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/>
<a NAME="E3:8"/>
<font color="Maroon">A</font> <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" target="_self">rng</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_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="E1:8_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:8_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" target="_self">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="8_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/>

<b>assume </b><a NAME="E1:8_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:8_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:8_1_1"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:8_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:8_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:8_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" target="_self">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>
<a NAME="E4:8"/><b>then </b>
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K1">meet</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/>
<b>hence </b><a NAME="E5:8"/>
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K1">meet</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/>


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