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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="measure1.html#M3" target="_self">sigma_Measure</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<a NAME="E1:65"/><i><font color="Green">E36</font></i>: 
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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="E2:65"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="measure1.html#R2" target="_self">is_measurable</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="measure1.html#R2" target="_self">is_measurable</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">b<sub>1</sub></font> <a href="measure1.html#R2" target="_self">is_measurable</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b<sub>2</sub></font> <a href="measure1.html#R2" target="_self">is_measurable</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="measure1.html#R2" target="_self">is_measurable</a> <font color="Maroon">c<sub>3</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:65_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="measure1.html#R2" target="_self">is_measurable</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#R2" target="_self">is_measurable</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:65_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="measure1.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E3:65_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="measure1.html#T19" target="_self">Th19</a>, <a class="ref" href="finsub_1.html#D1">FINSUB_1:def 1</a>, <a class="ref" href="measure1.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E4:65_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="measure1.html#R2" target="_self">is_measurable</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#R2" target="_self">is_measurable</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#R2" target="_self">is_measurable</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="measure1.html#D12" target="_self">Def12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:65"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="measure1.html#R2" target="_self">is_measurable</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="measure1.html#R2" target="_self">is_measurable</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="measure1.html#R2" target="_self">is_measurable</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="measure1.html#R2" target="_self">is_measurable</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">b<sub>1</sub></font> <a href="measure1.html#R2" target="_self">is_measurable</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b<sub>2</sub></font> <a href="measure1.html#R2" target="_self">is_measurable</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="measure1.html#R2" target="_self">is_measurable</a> <font color="Maroon">c<sub>3</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="measure1.html#E1:65"><i><font color="Green">E36</font></i></a>, <a class="ref" href="measure1.html#D12" target="_self">Def12</a></i>;<br/>


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