<?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#NM1">Field_Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">M</font> be    <a href="measure1.html#M1" target="_self">Measure</a> of <font color="Maroon">S</font>;<br/>





<a NAME="E1:29"/><i><font color="Green">E21</font></i>: 
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="prob_1.html#T10">PROB_1:10</a>, <a class="ref" href="prob_1.html#T11">PROB_1:11</a></i>;<br/>
<a NAME="E2:29"/>
for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="measure1.html#R1" target="_self">is_measurable</a> <font color="Maroon">M</font> &amp; <font color="Olive">B</font> <a href="measure1.html#R1" target="_self">is_measurable</a> <font color="Maroon">M</font> holds <br/>( <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Olive">A</font> <a href="measure1.html#R1" target="_self">is_measurable</a> <font color="Maroon">M</font> &amp; <font color="Olive">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">B</font> <a href="measure1.html#R1" target="_self">is_measurable</a> <font color="Maroon">M</font> &amp; <font color="Olive">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">B</font> <a href="measure1.html#R1" target="_self">is_measurable</a> <font color="Maroon">M</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:29_1"/>
( <font color="Maroon">A</font> <a href="measure1.html#R1" target="_self">is_measurable</a> <font color="Maroon">M</font> &amp; <font color="Maroon">B</font> <a href="measure1.html#R1" target="_self">is_measurable</a> <font color="Maroon">M</font> )
 ;<br/>

<a NAME="E2:29_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="measure1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E3:29_1"/><b>then </b>
( <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, , <a class="ref" href="finsub_1.html#D1">FINSUB_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:29_1"/>
( <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font> <a href="measure1.html#R1" target="_self">is_measurable</a> <font color="Maroon">M</font> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font> <a href="measure1.html#R1" target="_self">is_measurable</a> <font color="Maroon">M</font> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font> <a href="measure1.html#R1" target="_self">is_measurable</a> <font color="Maroon">M</font> )
 <b>by </b><i><a class="ref" href="measure1.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:29"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="measure1.html#R1" target="_self">is_measurable</a> <font color="Maroon">M</font> &amp; <font color="Maroon">X</font> <a href="measure1.html#R1" target="_self">is_measurable</a> <font color="Maroon">M</font> &amp; ( for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="measure1.html#R1" target="_self">is_measurable</a> <font color="Maroon">M</font> &amp; <font color="Olive">B</font> <a href="measure1.html#R1" target="_self">is_measurable</a> <font color="Maroon">M</font> holds <br/>( <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Olive">A</font> <a href="measure1.html#R1" target="_self">is_measurable</a> <font color="Maroon">M</font> &amp; <font color="Olive">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">B</font> <a href="measure1.html#R1" target="_self">is_measurable</a> <font color="Maroon">M</font> &amp; <font color="Olive">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">B</font> <a href="measure1.html#R1" target="_self">is_measurable</a> <font color="Maroon">M</font> ) ) )
 <b>by </b><i>, <a class="ref" href="measure1.html#T4" target="_self">Th4</a></i>;<br/>


</div>
