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<b>let </b><font color="Maroon" title="c1">Omega</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">Z</font> be   <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c1">Omega</font>;<br/>



<b>thus </b><a NAME="E1:80"/>
 <a href="prob_1.html#K11" title="PROB_1:func.11">sigma</a> <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_3.html#K14" title="PROB_3:func.14">monotoneclass</a> <font color="Maroon" title="c2">Z</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="PROB_3:func.14">monotoneclass</a> <font color="Maroon" title="c2">Z</font> is   <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c1">Omega</font>
 
<b>by </b><i><a class="ref" href="prob_3.html#T78" target="_self" title="PROB_3:th.78">Th78</a></i>;<br/>
<a NAME="E2:80_1"/><b>then </b><i><font color="Green" title="E68">A1</font></i>: 
 <a href="prob_3.html#K14" title="PROB_3:func.14">monotoneclass</a> <font color="Maroon" title="c2">Z</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font>
 
<b>by </b><i><a class="ref" href="prob_3.html#T75" target="_self" title="PROB_3:th.75">Th75</a></i>;<br/>
<a NAME="E3:80_1"/>
<font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_3.html#K14" title="PROB_3:func.14">monotoneclass</a> <font color="Maroon" title="c2">Z</font>
 
<b>by </b><i><a class="ref" href="prob_3.html#D15" target="_self" title="PROB_3:def.15">Def15</a></i>;<br/>
<b>hence </b><a NAME="E4:80_1"/>
 <a href="prob_1.html#K11" title="PROB_1:func.11">sigma</a> <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_3.html#K14" title="PROB_3:func.14">monotoneclass</a> <font color="Maroon" title="c2">Z</font>
 <b>by </b><i><a class="txt" href="prob_3.html#E2:80_1"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="prob_1.html#D14" title="PROB_1:def.14">PROB_1:def 14</a></i>;<br/>


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

<a NAME="E2:80"/><i><font color="Green" title="E68">A2</font></i>: 
 <a href="prob_1.html#K11" title="PROB_1:func.11">sigma</a> <font color="Maroon" title="c2">Z</font> is    <a href="prob_3.html#M2" title="PROB_3:mode.2">MonotoneClass</a> of <font color="Maroon" title="c1">Omega</font>
 
<b>by </b><i><a class="ref" href="prob_3.html#T75" target="_self" title="PROB_3:th.75">Th75</a></i>;<br/>
<a NAME="E3:80"/>
<font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K11" title="PROB_1:func.11">sigma</a> <font color="Maroon" title="c2">Z</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D14" title="PROB_1:def.14">PROB_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E4:80"/>
 <a href="prob_3.html#K14" title="PROB_3:func.14">monotoneclass</a> <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K11" title="PROB_1:func.11">sigma</a> <font color="Maroon" title="c2">Z</font>
 <b>by </b><i><a class="txt" href="prob_3.html#E2:80"><i><font color="Green" title="E68">A2</font></i></a>, <a class="ref" href="prob_3.html#D15" target="_self" title="PROB_3:def.15">Def15</a></i>;<br/>


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