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

<b>reconsider </b><font color="Maroon">S</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font></span><a href="tarski.html#K2">}</a></span> 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> <b>by </b><i><a class="ref" href="prob_1.html#T14">PROB_1:14</a></i>;<br/>
<b>take </b>
<font color="Maroon">S</font>
;<br/>

<b>thus </b><a NAME="E2:44_1_1"/>
not <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>thus </b><a NAME="E3:44_1_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>
  <i><font color="Red">:: according to </font></i><a class="ref" href="measure1.html#D3" target="_self">MEASURE1:def 3</a><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">A</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:44_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:44_1_1_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <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><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E3:44_1_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> implies <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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_1_1_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:44_1_1_1_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#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b><a NAME="E3:44_1_1_1_1"/>
<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><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:44_1_1_1"/>
<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><a class="txt" href="measure1.html#E2:44_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="measure1.html#E4:44_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>

<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>; <i><font color="Red">:: according to </font></i><a class="ref" href="measure1.html#D9" target="_self">MEASURE1:def 9</a><br/>

<b>assume </b><a NAME="E4:44_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E5:44_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> implies  <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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_1_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:44_1_1_2"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E3:44_1_1_2"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E4:44_1_1_2"/>
 <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><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:44_1_1"/>
( not <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> implies  <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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_1_1_3"/>
not <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:44_1_1_3"/><b>then </b>
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">M</font> holds <br/><font color="Olive">A</font> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E3:44_1_1_3"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/>
<a NAME="E4:44_1_1_3"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><a NAME="E5:44_1_1_3"/>
 <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><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:44_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><a class="txt" href="measure1.html#E4:44_1_1"><i><font color="Green">E25</font></i></a></i>;<br/>


</div>
