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<b>let </b><font color="Maroon">Omega</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Z</font> be   <a href="prob_1.html#NM1">Field_Subset</a> of <font color="Maroon">Omega</font>;<br/>



<b>thus </b><a NAME="E1:80"/>
 <a href="prob_1.html#K11">sigma</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="prob_3.html#K14" target="_self">monotoneclass</a> <font color="Maroon">Z</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<a NAME="E1:80_1"/>
 <a href="prob_3.html#K14" target="_self">monotoneclass</a> <font color="Maroon">Z</font> is   <a href="prob_1.html#NM1">Field_Subset</a> of <font color="Maroon">Omega</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:80_1"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="prob_3.html#K14" target="_self">monotoneclass</a> <font color="Maroon">Z</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:80_1"/>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="prob_3.html#K14" target="_self">monotoneclass</a> <font color="Maroon">Z</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:80_1"/>
 <a href="prob_1.html#K11">sigma</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="prob_3.html#K14" target="_self">monotoneclass</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="prob_3.html#D2" target="_self">Def2</a>, <a class="ref" href="prob_1.html#D14">PROB_1:def 14</a></i>;<br/>


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

<a NAME="E2:80"/><i><font color="Green">E52</font></i>: 
 <a href="prob_1.html#K11">sigma</a> <font color="Maroon">Z</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:80"/>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="prob_1.html#K11">sigma</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D14">PROB_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E4:80"/>
 <a href="prob_3.html#K14" target="_self">monotoneclass</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="prob_1.html#K11">sigma</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="prob_3.html#D2" target="_self">Def2</a>, </i>;<br/>


</div>
