<?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  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/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:47_1"/><i><font color="Green">E21</font></i>: 
( ( 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">M</font> being  <a href="measure1.html#NM3">N_Sub_set_fam</a> of <font color="Maroon">X</font>  st <font color="Olive">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> holds <br/> <a href="setfam_1.html#K6">meet</a> <font color="Olive">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) )
 ;<br/><a NAME="E2:47_1"/>
for <font color="Olive">M</font> being  <a href="measure1.html#NM3">N_Sub_set_fam</a> of <font color="Maroon">X</font>  st <font color="Olive">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> holds <br/> <a href="setfam_1.html#K5">union</a> <font color="Olive">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">M</font> be   <a href="measure1.html#NM3">N_Sub_set_fam</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:47_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:47_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">X</font> <a href="measure1.html#NK1" target="_self">\</a> <font color="Maroon">M</font> is   <a href="measure1.html#NM3">N_Sub_set_fam</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="measure1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E3:47_1_1"/>
<font color="Maroon">X</font> <a href="measure1.html#NK1" target="_self">\</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_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> ; <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="E1:47_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="measure1.html#NK1" target="_self">\</a> <font color="Maroon">M</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">y</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E3:47_1_1_1"/>
<font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E4:47_1_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="measure1.html#E2:47_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="measure1.html#E1:47_1_1_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E5:47_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
;<br/>
<b>hence </b><a NAME="E6:47_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:47_1_1"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="measure1.html#NK1" target="_self">\</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="measure1.html#E2:47_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="measure1.html#E2:47"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E5:47_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="measure1.html#NK1" target="_self">\</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="measure1.html#E2:47_1_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:47_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">S'</font> = <font color="Maroon">S</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="prob_1.html#V1">compl-closed</a> <a href="measure1.html#V2" target="_self">sigma-additive</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> <b>by </b><i>, , </i>;<br/><a NAME="E4:47_1"/>
<font color="Maroon">S'</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>
 ;<br/><b>hence </b><a NAME="E5:47_1"/>
<font color="Maroon">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:47"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:47"/>
for <font color="Olive">M</font> being  <a href="measure1.html#NM3">N_Sub_set_fam</a> of <font color="Maroon">X</font>  st <font color="Olive">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> holds <br/> <a href="setfam_1.html#K6">meet</a> <font color="Olive">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">M</font> be   <a href="measure1.html#NM3">N_Sub_set_fam</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:47_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:47_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">X</font> <a href="measure1.html#NK1" target="_self">\</a> <font color="Maroon">M</font> is   <a href="measure1.html#NM3">N_Sub_set_fam</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="measure1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E3:47_2"/>
<font color="Maroon">X</font> <a href="measure1.html#NK1" target="_self">\</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:47_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="measure1.html#NK1" target="_self">\</a> <font color="Maroon">M</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">y</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E3:47_2_1"/>
<font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E4:47_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="prob_1.html#D1">PROB_1:def 1</a>, </i>;<br/>
<b>hence </b><a NAME="E5:47_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:47_2"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="measure1.html#NK1" target="_self">\</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="txt" href="measure1.html#E2:47"><i><font color="Green">E27</font></i></a>, </i>;<br/>
<a NAME="E5:47_2"/><b>then </b>
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="measure1.html#NK1" target="_self">\</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:47_2"/>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">M</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="E4:47"/>
( ( 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">M</font> being  <a href="measure1.html#NM3">N_Sub_set_fam</a> of <font color="Maroon">X</font>  st <font color="Olive">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> holds <br/> <a href="setfam_1.html#K6">meet</a> <font color="Olive">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) )
 <b>by </b><i>, </i>;<br/>


</div>
