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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Z</font> being  <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c1">Omega</font> holds   <a href="prob_3.html#K14" title="PROB_3:func.14">monotoneclass</a> <font color="Olive" title="b1">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></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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></span><br/>



<a NAME="E1:79"/><i><font color="Green" title="E67">A1</font></i>: 
<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>then reconsider </b><font color="Maroon" title="c3">Z1</font> =  <a href="prob_3.html#K14" title="PROB_3:func.14">monotoneclass</a> <font color="Maroon" title="c2">Z</font> as  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">Omega</font> ;<br/>
<a NAME="E3:79"/><i><font color="Green" title="E68">A2</font></i>: 
for <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">x</font> where <font color="Olive" title="b3">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Olive" title="b1">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b3">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b3">x</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span>  holds <br/>for <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Y</font> holds <br/>( <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b3">z</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b3">z</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span>  implies for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> holds <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">z</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">z</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) )</span><br/>



<b>assume </b><a NAME="E1:79_1"/><i><font color="Green" title="E69">A3</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> holds <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">z</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">z</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> )</span><br/>

<b>thus </b><a NAME="E2:79_1"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> holds <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">z</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">z</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> implies ( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c6">z</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c6">z</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) )</span><br/>

<b>assume </b><a NAME="E1:79_1_1"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c6">z</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c6">z</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c7">z1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font><b> such that </b><br/><a NAME="E3:79_1_1"/><i><font color="Green" title="E70">A4</font></i>: 
( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c7">z1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c7">z1</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> )
 <b>by </b><i><a class="txt" href="#E1:79_1"><i><font color="Green" title="E69">A3</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:79_1_1"/>
( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c6">z</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c6">z</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> )
 <b>by </b><i><a class="txt" href="#E3:79_1_1"><i><font color="Green" title="E70">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div><b>end;</b></div>
<a NAME="E4:79"/><i><font color="Green" title="E69">A5</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font><br/> for <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">x</font> where <font color="Olive" title="b3">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Olive" title="b1">y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b3">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span>  holds <br/><font color="Olive" title="b2">Y</font> is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <font color="Maroon" title="c1">Omega</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">y</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b2">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b2">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span>  holds <br/><font color="Olive" title="b1">Y</font> is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <font color="Maroon" title="c1">Omega</font></span><br/><b>let </b><font color="Maroon" title="c5">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span>  implies <font color="Maroon" title="c5">Y</font> is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <font color="Maroon" title="c1">Omega</font> )</span><br/>



<b>assume </b><a NAME="E1:79_2"/><i><font color="Green" title="E70">A6</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">Y</font> is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <font color="Maroon" title="c1">Omega</font></span><br/>

<a NAME="E2:79_2"/><i><font color="Green" title="E71">A7</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">Omega</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:79_2_1"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 
<b>by </b><i><a class="txt" href="#E3:79"><i><font color="Green" title="E68">A2</font></i></a>, <a class="txt" href="#E1:79_2"><i><font color="Green" title="E70">A6</font></i></a></i>;<br/>
<a NAME="E2:79_2_1"/><b>then </b>
<font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Z1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:79_2_1"/>
<font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">Omega</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:79_2"/>
( <font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">Omega</font> &amp; ( for <font color="Olive" title="b1">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">Omega</font>  st <font color="Olive" title="b1">A1</font> is <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font> holds <br/> <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b1">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:79_2_2"/>
for <font color="Olive" title="b1">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">Omega</font>  st <font color="Olive" title="b1">A1</font> is <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font> holds <br/> <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b1">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">A1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">Omega</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">A1</font> is <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font> implies  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c6">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:79_2_2_1"/><i><font color="Green" title="E72">A8</font></i>: 
<font color="Maroon" title="c6">A1</font> is <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a>
 <b>and </b><br/><a NAME="E2:79_2_2_1"/><i><font color="Green" title="E73">A9</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c6">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font></span><br/>

<a NAME="E3:79_2_2_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c6">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font></span><br/>

<a NAME="E1:79_2_2_1_1"/>
<font color="Maroon" title="c6">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">A1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T52" title="NAT_1:th.52">NAT_1:52</a></i>;<br/>
<a NAME="E2:79_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="txt" href="#E2:79_2_2_1"><i><font color="Green" title="E73">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:79_2_2_1_1"/>
<font color="Maroon" title="c6">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 <b>by </b><i><a class="txt" href="#E3:79"><i><font color="Green" title="E68">A2</font></i></a>, <a class="txt" href="#E1:79_2"><i><font color="Green" title="E70">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:79_2_2_1"/><b>then </b><i><font color="Green" title="E74">B10</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Z1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T53" title="NAT_1:th.53">NAT_1:53</a></i>;<br/>
<a NAME="E5:79_2_2_1"/><i><font color="Green" title="E75">A11</font></i>: 
<font color="Maroon" title="c6">A1</font> is <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a>
 
<b>by </b><i><a class="txt" href="#E1:79_2_2_1"><i><font color="Green" title="E72">A8</font></i></a>, <a class="ref" href="setlim_1.html#T65" title="SETLIM_1:th.65">SETLIM_1:65</a></i>;<br/>
<a NAME="E6:79_2_2_1"/><i><font color="Green" title="E76">A12</font></i>: 
<font color="Maroon" title="c6">A1</font> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Maroon" title="c4">y</font> is <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a>
 
<b>by </b><i><a class="txt" href="#E1:79_2_2_1"><i><font color="Green" title="E72">A8</font></i></a>, <a class="ref" href="setlim_2.html#T32" title="SETLIM_2:th.32">SETLIM_2:32</a></i>;<br/>
<a NAME="E7:79_2_2_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font></span><br/>

<a NAME="E1:79_2_2_1_2"/><i><font color="Green" title="E77">A13</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:79_2_2_1_2"/>
<font color="Maroon" title="c6">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">A1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T52" title="NAT_1:th.52">NAT_1:52</a></i>;<br/>
<a NAME="E3:79_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c6">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="txt" href="#E2:79_2_2_1"><i><font color="Green" title="E73">A9</font></i></a></i>;<br/>
<a NAME="E4:79_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 
<b>by </b><i><a class="txt" href="#E3:79"><i><font color="Green" title="E68">A2</font></i></a>, <a class="txt" href="#E1:79_2"><i><font color="Green" title="E70">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:79_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 <b>by </b><i><a class="txt" href="#E1:79_2_2_1_2"><i><font color="Green" title="E77">A13</font></i></a>, <a class="ref" href="setlim_2.html#D8" title="SETLIM_2:def.8">SETLIM_2:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:79_2_2_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Z1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T53" title="NAT_1:th.53">NAT_1:53</a></i>;<br/>
<a NAME="E9:79_2_2_1"/><b>then </b><i><font color="Green" title="E77">A14</font></i>: 
 <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 
<b>by </b><i><a class="txt" href="#E6:79_2_2_1"><i><font color="Green" title="E76">A12</font></i></a>, <a class="ref" href="prob_3.html#T74" target="_self" title="PROB_3:th.74">Th74</a></i>;<br/>
<a NAME="E10:79_2_2_1"/><i><font color="Green" title="E78">A15</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c6">A1</font> is <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a>
 
<b>by </b><i><a class="txt" href="#E1:79_2_2_1"><i><font color="Green" title="E72">A8</font></i></a>, <a class="ref" href="setlim_2.html#T29" title="SETLIM_2:th.29">SETLIM_2:29</a></i>;<br/>
<a NAME="E11:79_2_2_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font></span><br/>

<a NAME="E1:79_2_2_1_3"/><i><font color="Green" title="E79">A16</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:79_2_2_1_3"/>
<font color="Maroon" title="c6">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">A1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T52" title="NAT_1:th.52">NAT_1:52</a></i>;<br/>
<a NAME="E3:79_2_2_1_3"/><b>then </b>
<font color="Maroon" title="c6">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="txt" href="#E2:79_2_2_1"><i><font color="Green" title="E73">A9</font></i></a></i>;<br/>
<a NAME="E4:79_2_2_1_3"/><b>then </b>
<font color="Maroon" title="c4">y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 
<b>by </b><i><a class="txt" href="#E3:79"><i><font color="Green" title="E68">A2</font></i></a>, <a class="txt" href="#E1:79_2"><i><font color="Green" title="E70">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:79_2_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 <b>by </b><i><a class="txt" href="#E1:79_2_2_1_3"><i><font color="Green" title="E79">A16</font></i></a>, <a class="ref" href="setlim_2.html#D7" title="SETLIM_2:def.7">SETLIM_2:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E12:79_2_2_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Z1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T53" title="NAT_1:th.53">NAT_1:53</a></i>;<br/>
<a NAME="E13:79_2_2_1"/><b>then </b><i><font color="Green" title="E79">A17</font></i>: 
 <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 
<b>by </b><i><a class="txt" href="#E10:79_2_2_1"><i><font color="Green" title="E78">A15</font></i></a>, <a class="ref" href="prob_3.html#T74" target="_self" title="PROB_3:th.74">Th74</a></i>;<br/>
<a NAME="E14:79_2_2_1"/><i><font color="Green" title="E80">A18</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c6">A1</font> is <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a>
 
<b>by </b><i><a class="txt" href="#E1:79_2_2_1"><i><font color="Green" title="E72">A8</font></i></a>, <a class="ref" href="setlim_2.html#T23" title="SETLIM_2:th.23">SETLIM_2:23</a></i>;<br/>
<a NAME="E15:79_2_2_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font></span><br/>

<a NAME="E1:79_2_2_1_4"/><i><font color="Green" title="E81">A19</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:79_2_2_1_4"/>
<font color="Maroon" title="c6">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">A1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T52" title="NAT_1:th.52">NAT_1:52</a></i>;<br/>
<a NAME="E3:79_2_2_1_4"/><b>then </b>
<font color="Maroon" title="c6">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="txt" href="#E2:79_2_2_1"><i><font color="Green" title="E73">A9</font></i></a></i>;<br/>
<a NAME="E4:79_2_2_1_4"/><b>then </b>
<font color="Maroon" title="c4">y</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 
<b>by </b><i><a class="txt" href="#E3:79"><i><font color="Green" title="E68">A2</font></i></a>, <a class="txt" href="#E1:79_2"><i><font color="Green" title="E70">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:79_2_2_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 <b>by </b><i><a class="txt" href="#E1:79_2_2_1_4"><i><font color="Green" title="E81">A19</font></i></a>, <a class="ref" href="setlim_2.html#D5" title="SETLIM_2:def.5">SETLIM_2:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E16:79_2_2_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Z1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T53" title="NAT_1:th.53">NAT_1:53</a></i>;<br/>
<a NAME="E17:79_2_2_1"/><b>then </b><i><font color="Green" title="E81">B</font></i>: 
 <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 
<b>by </b><i><a class="txt" href="#E14:79_2_2_1"><i><font color="Green" title="E80">A18</font></i></a>, <a class="ref" href="prob_3.html#T74" target="_self" title="PROB_3:th.74">Th74</a></i>;<br/>
<a NAME="E18:79_2_2_1"/><i><font color="Green" title="E82">B1</font></i>: 
 <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c6">A1</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font>
 
<b>by </b><i><a class="txt" href="#E1:79_2_2_1"><i><font color="Green" title="E72">A8</font></i></a>, <a class="ref" href="prob_3.html#T74" target="_self" title="PROB_3:th.74">Th74</a>, <a class="txt" href="#E4:79_2_2_1"><i><font color="Green" title="E74">B10</font></i></a></i>;<br/>
<a NAME="E19:79_2_2_1"/>
( <span class="p1">(<span class="default"><a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> )
 
<b>by </b><i><a class="txt" href="#E5:79_2_2_1"><i><font color="Green" title="E75">A11</font></i></a>, <a class="txt" href="#E9:79_2_2_1"><i><font color="Green" title="E77">A14</font></i></a>, <a class="txt" href="#E13:79_2_2_1"><i><font color="Green" title="E79">A17</font></i></a>, <a class="ref" href="setlim_2.html#T92" title="SETLIM_2:th.92">SETLIM_2:92</a>, <a class="ref" href="setlim_2.html#T94" title="SETLIM_2:th.94">SETLIM_2:94</a>, <a class="ref" href="setlim_2.html#T95" title="SETLIM_2:th.95">SETLIM_2:95</a>, <a class="txt" href="#E17:79_2_2_1"><i><font color="Green" title="E81">B</font></i></a></i>;<br/>
<b>hence </b><a NAME="E20:79_2_2_1"/>
 <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c6">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="#E1:79_2"><i><font color="Green" title="E70">A6</font></i></a>, <a class="txt" href="#E18:79_2_2_1"><i><font color="Green" title="E82">B1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:79_2_2"/>
( <font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">Omega</font> &amp; ( for <font color="Olive" title="b1">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">Omega</font>  st <font color="Olive" title="b1">A1</font> is <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font> holds <br/> <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b1">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> ) )
 <b>by </b><i><a class="txt" href="#E2:79_2"><i><font color="Green" title="E71">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:79_2"/>
<font color="Maroon" title="c5">Y</font> is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <font color="Maroon" title="c1">Omega</font>
 <b>by </b><i><a class="ref" href="prob_3.html#T74" target="_self" title="PROB_3:th.74">Th74</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:79"/><i><font color="Green" title="E70">A20</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Z</font><br/> for <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">x</font> where <font color="Olive" title="b3">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Olive" title="b1">y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b3">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span>  holds <br/><font color="Maroon" title="c3">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">y</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Z</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b2">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b2">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span>  holds <br/><font color="Maroon" title="c3">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font></span><br/><b>let </b><font color="Maroon" title="c5">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span>  implies <font color="Maroon" title="c3">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font> )</span><br/>



<b>assume </b><a NAME="E1:79_3"/><i><font color="Green" title="E71">A21</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font></span><br/>

<a NAME="E2:79_3"/><i><font color="Green" title="E72">A22</font></i>: 
<font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">z</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> or <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> )</span><br/>

<b>assume </b><a NAME="E1:79_3_1"/><i><font color="Green" title="E73">A23</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font></span><br/>

<a NAME="E2:79_3_1"/><b>then </b><i><font color="Green" title="E74">A24</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T12" title="PROB_1:th.12">PROB_1:12</a></i>;<br/>
<a NAME="E3:79_3_1"/><i><font color="Green" title="E75">A25</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 
<b>by </b><i><a class="txt" href="#E1:79_3_1"><i><font color="Green" title="E73">A23</font></i></a>, <a class="ref" href="prob_1.html#T12" title="PROB_1:th.12">PROB_1:12</a></i>;<br/>
<a NAME="E4:79_3_1"/>
<font color="Maroon" title="c6">z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 
<b>by </b><i><a class="txt" href="#E1:79_3_1"><i><font color="Green" title="E73">A23</font></i></a>, <a class="ref" href="finsub_1.html#D2" title="FINSUB_1:def.2">FINSUB_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:79_3_1"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="#E1:79_3"><i><font color="Green" title="E71">A21</font></i></a>, <a class="txt" href="#E1:79_3_1"><i><font color="Green" title="E73">A23</font></i></a>, <a class="txt" href="#E2:79_3_1"><i><font color="Green" title="E74">A24</font></i></a>, <a class="txt" href="#E3:79_3_1"><i><font color="Green" title="E75">A25</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:79_3"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 
;<br/>
<a NAME="E4:79_3"/><b>then </b>
<font color="Maroon" title="c5">Y</font> is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <font color="Maroon" title="c1">Omega</font>
 
<b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="#E4:79"><i><font color="Green" title="E69">A5</font></i></a>, <a class="txt" href="#E1:79_3"><i><font color="Green" title="E71">A21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:79_3"/>
<font color="Maroon" title="c3">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="#E2:79_3"><i><font color="Green" title="E72">A22</font></i></a>, <a class="ref" href="prob_3.html#D15" target="_self" title="PROB_3:def.15">Def15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:79"/><i><font color="Green" title="E71">A26</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font><br/> for <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">x</font> where <font color="Olive" title="b3">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Olive" title="b1">y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b3">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span>  holds <br/><font color="Maroon" title="c3">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">y</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b2">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b2">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span>  holds <br/><font color="Maroon" title="c3">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font></span><br/><b>let </b><font color="Maroon" title="c5">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span>  implies <font color="Maroon" title="c3">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font> )</span><br/>



<b>assume </b><a NAME="E1:79_4"/><i><font color="Green" title="E72">A27</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font></span><br/>

<a NAME="E2:79_4"/><i><font color="Green" title="E73">A28</font></i>: 
<font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">z</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> or <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> )</span><br/>

<b>assume </b><a NAME="E1:79_4_1"/><i><font color="Green" title="E74">A29</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font></span><br/>

<b>set </b><font color="Maroon" title="c7">Y1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Maroon" title="c6">z</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span> ;<br/>
<a NAME="E2:79_4_1"/><i><font color="Green" title="E75">A30</font></i>: 
<font color="Maroon" title="c3">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Maroon" title="c6">z</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E5:79"><i><font color="Green" title="E70">A20</font></i></a>, <a class="txt" href="#E1:79_4_1"><i><font color="Green" title="E74">A29</font></i></a></i>;<br/>
<a NAME="E3:79_4_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 
;<br/>
<a NAME="E4:79_4_1"/><b>then </b>
( <font color="Maroon" title="c6">z</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c6">z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> )
 
<b>by </b><i><a class="txt" href="#E3:79"><i><font color="Green" title="E68">A2</font></i></a>, <a class="txt" href="#E2:79_4_1"><i><font color="Green" title="E75">A30</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:79_4_1"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="#E1:79_4"><i><font color="Green" title="E72">A27</font></i></a>, <a class="txt" href="#E1:79_4_1"><i><font color="Green" title="E74">A29</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:79_4"/>
<font color="Maroon" title="c5">Y</font> is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <font color="Maroon" title="c1">Omega</font>
 
<b>by </b><i><a class="txt" href="#E4:79"><i><font color="Green" title="E69">A5</font></i></a>, <a class="txt" href="#E1:79_4"><i><font color="Green" title="E72">A27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:79_4"/>
<font color="Maroon" title="c3">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="#E2:79_4"><i><font color="Green" title="E73">A28</font></i></a>, <a class="ref" href="prob_3.html#D15" target="_self" title="PROB_3:def.15">Def15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:79"/><i><font color="Green" title="E72">A31</font></i>: 
<font color="Maroon" title="c3">Z1</font> is <a href="finsub_1.html#V2" title="FINSUB_1:attr.2">cap-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> implies <font color="Maroon" title="c4">y</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> )</span><br/><b>assume </b><a NAME="E1:79_5_1"/><i><font color="Green" title="E73">A32</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">y</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font></span><br/><b>set </b><font color="Maroon" title="c6">Y</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span> ;<br/><a NAME="E2:79_5_1"/>
<font color="Maroon" title="c3">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E6:79"><i><font color="Green" title="E71">A26</font></i></a>, <a class="txt" href="#E1:79_5_1"><i><font color="Green" title="E73">A32</font></i></a></i>;<br/><b>hence </b><a NAME="E3:79_5_1"/>
<font color="Maroon" title="c4">y</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 <b>by </b><i><a class="txt" href="#E3:79"><i><font color="Green" title="E68">A2</font></i></a>, <a class="txt" href="#E1:79_5_1"><i><font color="Green" title="E73">A32</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:79_5"/>
<font color="Maroon" title="c3">Z1</font> is <a href="finsub_1.html#V2" title="FINSUB_1:attr.2">cap-closed</a>
 <b>by </b><i><a class="ref" href="finsub_1.html#D2" title="FINSUB_1:def.2">FINSUB_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:79"/>
for <font color="Olive" title="b1">y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Omega</font>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> holds <br/><font color="Olive" title="b1">y</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">Z1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Omega</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> implies <font color="Maroon" title="c4">y</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">Z1</font> )</span><br/>

<b>assume </b><a NAME="E1:79_6"/><i><font color="Green" title="E73">A33</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">y</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">Z1</font></span><br/>

<b>set </b><font color="Maroon" title="c5">Y</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span> ;<br/>
<a NAME="E2:79_6"/>
<font color="Maroon" title="c1">Omega</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T11" title="PROB_1:th.11">PROB_1:11</a></i>;<br/>
<a NAME="E3:79_6"/><b>then </b><i><font color="Green" title="E74">A34</font></i>: 
<font color="Maroon" title="c1">Omega</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 
<b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E67">A1</font></i></a></i>;<br/>
<a NAME="E4:79_6"/>
<font color="Maroon" title="c3">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> : ( <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Olive" title="b1">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E6:79"><i><font color="Green" title="E71">A26</font></i></a>, <a class="txt" href="#E1:79_6"><i><font color="Green" title="E73">A33</font></i></a></i>;<br/>
<a NAME="E5:79_6"/><b>then </b>
<font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z1</font>
 
<b>by </b><i><a class="txt" href="#E3:79"><i><font color="Green" title="E68">A2</font></i></a>, <a class="txt" href="#E3:79_6"><i><font color="Green" title="E74">A34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:79_6"/>
<font color="Maroon" title="c4">y</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">Z1</font>
 <b>by </b><i><a class="ref" href="subset_1.html#D5" title="SUBSET_1:def.5">SUBSET_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:79"/>
 <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="txt" href="#E7:79"><i><font color="Green" title="E72">A31</font></i></a>, <a class="ref" href="prob_1.html#D1" title="PROB_1:def.1">PROB_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
