<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:44_1"/><i><font color="Green" title="E24">A1</font></i>: 
( ( for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> holds <br/><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> ) &amp; ( for <font color="Olive" title="b1">M</font> being  <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> ) )
 ;<br/><a NAME="E2:44_1"/>
for <font color="Olive" title="b1">M</font> being  <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> holds <br/> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E1:44_1_1"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 ;<br/>

<a NAME="E2:44_1_1"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c3">M</font> is   <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="measure1.html#T42" target="_self" title="MEASURE1:th.42">Th42</a></i>;<br/>
<a NAME="E3:44_1_1"/>
<font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c3">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:44_1_1_1"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c3">M</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">B</font> = <font color="Maroon" title="c4">y</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:44_1_1_1"/>
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="measure1.html#E1:44_1_1_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E4:44_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="measure1.html#E1:44_1"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="measure1.html#E1:44_1_1"><i><font color="Green" title="E25">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:44_1_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:44_1_1"/><b>then </b>
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="measure1.html#E1:44_1"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="measure1.html#E2:44_1_1"><i><font color="Green" title="E26">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:44_1_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="measure1.html#T15" target="_self" title="MEASURE1:th.15">Th15</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c3">S'</font> = <font color="Maroon" title="c2">S</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="measure1.html#V2" title="MEASURE1:attr.2">sigma-additive</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="measure1.html#E1:44_1"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="measure1.html#D3" target="_self" title="MEASURE1:def.3">Def3</a>, <a class="ref" href="measure1.html#D9" target="_self" title="MEASURE1:def.9">Def9</a></i>;<br/><a NAME="E4:44_1"/>
<font color="Maroon" title="c3">S'</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>
 ;<br/><b>hence </b><a NAME="E5:44_1"/>
<font color="Maroon" title="c2">S</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:44"/><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c2">S</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E3:44"/>
for <font color="Olive" title="b1">M</font> being  <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E1:44_2"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 ;<br/>

<a NAME="E2:44_2"/><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c3">M</font> is   <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="measure1.html#T42" target="_self" title="MEASURE1:th.42">Th42</a></i>;<br/>
<a NAME="E3:44_2"/>
<font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c3">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:44_2_1"/><i><font color="Green" title="E27">A8</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c3">M</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">B</font> = <font color="Maroon" title="c4">y</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:44_2_1"/>
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="measure1.html#E1:44_2_1"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E4:44_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="measure1.html#E2:44"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="measure1.html#E1:44_2"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="prob_1.html#D1" title="PROB_1:def.1">PROB_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:44_2_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:44_2"/><b>then </b>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="measure1.html#E2:44"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="measure1.html#E2:44_2"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="measure1.html#D9" target="_self" title="MEASURE1:def.9">Def9</a></i>;<br/>
<a NAME="E5:44_2"/><b>then </b>
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="measure1.html#E2:44"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="measure1.html#D3" target="_self" title="MEASURE1:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E6:44_2"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="measure1.html#T15" target="_self" title="MEASURE1:th.15">Th15</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:44"/>
( ( for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> holds <br/><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> ) &amp; ( for <font color="Olive" title="b1">M</font> being  <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> ) )
 <b>by </b><i><a class="txt" href="measure1.html#E2:44"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="measure1.html#D3" target="_self" title="MEASURE1:def.3">Def3</a></i>;<br/>


</div>
