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<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">M</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">M</font> is   <a href="prob_1.html#NM1">Field_Subset</a> of <font color="Maroon">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon">S</font> = <font color="Maroon">M</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/><b>take </b><font color="Maroon">S</font> = <font color="Maroon">S</font>;<br/><b>thus </b><a NAME="E3:15_1"/>
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 ;<br/><b>thus </b><a NAME="E4:15_1"/>
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/>( <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</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="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Olive">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">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/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">S</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font><b> such that </b><a NAME="E2:15"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> &amp; ( for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/>( <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</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="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Olive">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) ) ) )
 ;<br/>

<a NAME="E3:15"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">S</font> is <a href="finsub_1.html#V1">cup-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="finsub_1.html#D1">FINSUB_1:def 1</a><br/>

<b>assume </b><a NAME="E1:15_2"/>
( <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> )
 ;<br/>

<b>hence </b><a NAME="E2:15_2"/>
<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>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:15"/>
<font color="Maroon">S</font> is <a href="prob_1.html#V1">compl-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="measure1.html#D3" target="_self">MEASURE1:def 3</a><br/>

<b>assume </b><a NAME="E1:15_3"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>hence </b><a NAME="E2:15_3"/>
<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>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:15"/>
<font color="Maroon">M</font> is   <a href="prob_1.html#NM1">Field_Subset</a> of <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/>


</div>
