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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </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#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">C</font> being   <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font> st <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> );<br/>
<b>consider </b><font color="Maroon">D</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:22_1_1"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> iff ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:22_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:22_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>hence </b><a NAME="E2:22_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:22_1_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> iff  ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">C</font> being   <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font> st <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:22_1_1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> iff ( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">C</font> being   <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font> st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> ) ) ) )
 
<b>by </b><i/>;<br/>
<a NAME="E2:22_1_1_2"/>
(  ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">C</font> being   <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> ) implies <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1_1_2_1"/><i><font color="Green">E30</font></i>: 
 ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">C</font> being   <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_1_1_2_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">C</font> being   <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> )
 ;<br/>
<a NAME="E4:22_1_1_2_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="measure3.html#E2:22_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E5:22_1_1_2_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:22_1_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> iff  ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">C</font> being   <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22_1_1"/><i><font color="Green">E40</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E6:22_1_1"/>
 ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> &amp; <font color="Maroon">M</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:22_1_1_3"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T10">PROB_1:10</a></i>;<br/>
<b>then consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_1_1_3"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 ;<br/>
<b>take </b>
<font color="Maroon">B</font>
;<br/>

<b>thus </b><a NAME="E4:22_1_1_3"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">M</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, <a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  is    <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:22_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E9:22_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#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/> ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">C</font> being   <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font> st <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_4"><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="E1:22_1_1_4"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:22_1_1_4"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T10">PROB_1:10</a></i>;<br/>
<b>then consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:22_1_1_4"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">C</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<a NAME="E6:22_1_1_4"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="measure3.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E7:22_1_1_4"/>
 ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">C</font> being   <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="ref" href="measure3.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</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> <b>by </b><i>, <a class="txt" href="measure3.html#E1"><i><font color="Green">Lemma15</font></i></a>, <a class="txt" href="measure3.html#E2:22_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">D</font>
;<br/>

<b>thus </b><a NAME="E11:22_1_1"/>
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> iff  ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">C</font> being   <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font> st <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> ) )
 <b>by </b><i/>;<br/>


</div>
