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<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"/><span class="lab"><font color="Green" title="E66">A1</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#D15" target="_self" title="PROB_3:def.15">Def15</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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"/><span class="lab"><font color="Green" title="E67">A2</font></span>: 
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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>



<span class="kw">assume </span><a NAME="E1:79_1"/><span class="lab"><font color="Green" title="E68">A3</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">thus </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<a NAME="E2:79_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">z1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Z1</font> st <br/>( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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">z1</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">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="Olive" title="b1">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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_1"><span class="lab"><font color="Green" title="E68">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:79_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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E4:79"/><span class="lab"><font color="Green" title="E68">A4</font></span>: 
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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>



<span class="kw">assume </span><a NAME="E1:79_2"/><span class="lab"><font color="Green" title="E69">A5</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"/><span class="lab"><font color="Green" title="E70">A6</font></span>: 
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#K2" title="RELSET_1:func.2">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_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K2" title="RELSET_1:func.2">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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:79_2_1"/><span class="lab"><font color="Green" title="E71">A7</font></span>: 
<font color="Maroon" title="c6">A1</font> is  <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a> 
 <span class="kw">and </span><br/><a NAME="E2:79_2_1"/><span class="lab"><font color="Green" title="E72">A8</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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_1"/><span class="lab"><font color="Green" title="E73">A9</font></span>: 
<font color="Maroon" title="c6">A1</font> is  <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_2_1"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="ref" href="setlim_1.html#T65" title="SETLIM_1:th.65">SETLIM_1:65</a></span>;<br/>
<a NAME="E4:79_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_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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_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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T52" title="NAT_1:th.52">NAT_1:52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:79_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E1:79_2"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E2:79_2_1"><span class="lab"><font color="Green" title="E72">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:79_2_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T53" title="NAT_1:th.53">NAT_1:53</a></span>;<br/>
<a NAME="E6:79_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A10</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_2_1"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="ref" href="prob_3.html#T74" target="_self" title="PROB_3:th.74">Th74</a></span>;<br/>
<a NAME="E7:79_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_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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_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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T52" title="NAT_1:th.52">NAT_1:52</a></span>;<br/>
<a NAME="E2:79_2_1_2"/><span class="kw">then </span>
( <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>  &amp; <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E1:79_2"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E2:79_2_1"><span class="lab"><font color="Green" title="E72">A8</font></span></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:79_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_2.html#D8" title="SETLIM_2:def.8">SETLIM_2:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:79_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A11</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T53" title="NAT_1:th.53">NAT_1:53</a></span>;<br/>
<a NAME="E9:79_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_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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_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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T52" title="NAT_1:th.52">NAT_1:52</a></span>;<br/>
<a NAME="E2:79_2_1_3"/><span class="kw">then </span>
( <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>  &amp; <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E1:79_2"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E2:79_2_1"><span class="lab"><font color="Green" title="E72">A8</font></span></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:79_2_1_3"/>
<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 class="kw">by</span> <span class="lab"><a class="ref" href="setlim_2.html#D5" title="SETLIM_2:def.5">SETLIM_2:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:79_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A12</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T53" title="NAT_1:th.53">NAT_1:53</a></span>;<br/>
<a NAME="E11:79_2_1"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_2_1"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="ref" href="setlim_2.html#T23" title="SETLIM_2:th.23">SETLIM_2:23</a></span>;<br/>
<a NAME="E12:79_2_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:79_2_1"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="ref" href="prob_3.html#T74" target="_self" title="PROB_3:th.74">Th74</a></span>;<br/>
<a NAME="E13:79_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A13</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79_2_1"><span class="lab"><font color="Green" title="E73">A9</font></span></a>, <a class="ref" href="setlim_2.html#T92" title="SETLIM_2:th.92">SETLIM_2:92</a></span>;<br/>
<a NAME="E14:79_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_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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_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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T52" title="NAT_1:th.52">NAT_1:52</a></span>;<br/>
<a NAME="E2:79_2_1_4"/><span class="kw">then </span>
( <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>  &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"><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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E1:79_2"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E2:79_2_1"><span class="lab"><font color="Green" title="E72">A8</font></span></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:79_2_1_4"/>
<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 class="kw">by</span> <span class="lab"><a class="ref" href="setlim_2.html#D7" title="SETLIM_2:def.7">SETLIM_2:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:79_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A14</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T53" title="NAT_1:th.53">NAT_1:53</a></span>;<br/>
<a NAME="E16:79_2_1"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_2_1"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="ref" href="setlim_2.html#T29" title="SETLIM_2:th.29">SETLIM_2:29</a></span>;<br/>
<a NAME="E17:79_2_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:79_2_1"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="ref" href="prob_3.html#T74" target="_self" title="PROB_3:th.74">Th74</a></span>;<br/>
<a NAME="E18:79_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A15</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79_2_1"><span class="lab"><font color="Green" title="E73">A9</font></span></a>, <a class="ref" href="setlim_2.html#T94" title="SETLIM_2:th.94">SETLIM_2:94</a></span>;<br/>
<a NAME="E19:79_2_1"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_2_1"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="ref" href="setlim_2.html#T32" title="SETLIM_2:th.32">SETLIM_2:32</a></span>;<br/>
<a NAME="E20:79_2_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:79_2_1"><span class="lab"><font color="Green" title="E75">A11</font></span></a>, <a class="ref" href="prob_3.html#T74" target="_self" title="PROB_3:th.74">Th74</a></span>;<br/>
<a NAME="E21:79_2_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79_2_1"><span class="lab"><font color="Green" title="E73">A9</font></span></a>, <a class="ref" href="setlim_2.html#T95" title="SETLIM_2:th.95">SETLIM_2:95</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:79_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_2"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E6:79_2_1"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="txt" href="#E18:79_2_1"><span class="lab"><font color="Green" title="E79">A15</font></span></a>, <a class="txt" href="#E13:79_2_1"><span class="lab"><font color="Green" title="E77">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:79_2"/>
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E1:79_2"><span class="lab"><font color="Green" title="E69">A5</font></span></a></span>;<br/>
<a NAME="E4:79_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:79_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79_2"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="ref" href="prob_3.html#T74" target="_self" title="PROB_3:th.74">Th74</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:79"/><span class="lab"><font color="Green" title="E69">A16</font></span>: 
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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>



<span class="kw">assume </span><a NAME="E1:79_3"/><span class="lab"><font color="Green" title="E70">A17</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"/><span class="lab"><font color="Green" title="E71">A18</font></span>: 
<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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:79_3_1"/><span class="lab"><font color="Green" title="E72">A19</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A20</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D2" title="FINSUB_1:def.2">FINSUB_1:def 2</a></span>;<br/>
<a NAME="E3:79_3_1"/>
( <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> &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="c2">Z</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_3_1"><span class="lab"><font color="Green" title="E72">A19</font></span></a>, <a class="ref" href="prob_1.html#T12" title="PROB_1:th.12">PROB_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:79_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="txt" href="#E1:79_3"><span class="lab"><font color="Green" title="E70">A17</font></span></a>, <a class="txt" href="#E1:79_3_1"><span class="lab"><font color="Green" title="E72">A19</font></span></a>, <a class="txt" href="#E2:79_3_1"><span class="lab"><font color="Green" title="E73">A20</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></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"/><span class="kw">then </span>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="txt" href="#E4:79"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E1:79_3"><span class="lab"><font color="Green" title="E70">A17</font></span></a></span>;<br/>
<span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79_3"><span class="lab"><font color="Green" title="E71">A18</font></span></a>, <a class="ref" href="prob_3.html#D15" target="_self" title="PROB_3:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:79"/><span class="lab"><font color="Green" title="E70">A21</font></span>: 
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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>



<span class="kw">assume </span><a NAME="E1:79_4"/><span class="lab"><font color="Green" title="E71">A22</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"/><span class="lab"><font color="Green" title="E72">A23</font></span>: 
<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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">set </span><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/>
<span class="kw">assume </span><a NAME="E1:79_4_1"/><span class="lab"><font color="Green" title="E73">A24</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A25</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:79"><span class="lab"><font color="Green" title="E69">A16</font></span></a></span>;<br/>
<a NAME="E3:79_4_1"/><span class="lab"><font color="Green" title="E75">A26</font></span>: 
<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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A27</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E2:79_4_1"><span class="lab"><font color="Green" title="E74">A25</font></span></a></span>;<br/>
<a NAME="E5:79_4_1"/>
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E2:79_4_1"><span class="lab"><font color="Green" title="E74">A25</font></span></a>, <a class="txt" href="#E3:79_4_1"><span class="lab"><font color="Green" title="E75">A26</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:79_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="txt" href="#E1:79_4"><span class="lab"><font color="Green" title="E71">A22</font></span></a>, <a class="txt" href="#E1:79_4_1"><span class="lab"><font color="Green" title="E73">A24</font></span></a>, <a class="txt" href="#E4:79_4_1"><span class="lab"><font color="Green" title="E76">A27</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E1:79_4"><span class="lab"><font color="Green" title="E71">A22</font></span></a></span>;<br/>
<span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79_4"><span class="lab"><font color="Green" title="E72">A23</font></span></a>, <a class="ref" href="prob_3.html#D15" target="_self" title="PROB_3:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:79"/><span class="lab"><font color="Green" title="E71">A28</font></span>: 
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_5"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:79_5"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T11" title="PROB_1:th.11">PROB_1:11</a></span>;<br/>
<a NAME="E2:79_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A29</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E66">A1</font></span></a></span>;<br/>
<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E3:79_5"/><span class="lab"><font color="Green" title="E73">A30</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">set </span><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="E4:79_5"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:79"><span class="lab"><font color="Green" title="E70">A21</font></span></a>, <a class="txt" href="#E3:79_5"><span class="lab"><font color="Green" title="E73">A30</font></span></a></span>;<br/>
<a NAME="E5:79_5"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E2:79_5"><span class="lab"><font color="Green" title="E72">A29</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:79_5"/>
<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 class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#D5" title="SUBSET_1:def.5">SUBSET_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:79_6"/><span class="lab"><font color="Green" title="E72">A31</font></span>: 
<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 class="kw">and </span><br/><a NAME="E2:79_6"/><span class="lab"><font color="Green" title="E73">A32</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">set </span><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="E3:79_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#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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:79"><span class="lab"><font color="Green" title="E70">A21</font></span></a>, <a class="txt" href="#E1:79_6"><span class="lab"><font color="Green" title="E72">A31</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:79_6"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E2:79_6"><span class="lab"><font color="Green" title="E73">A32</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:79"><span class="lab"><font color="Green" title="E71">A28</font></span></a>, <a class="ref" href="finsub_1.html#D2" title="FINSUB_1:def.2">FINSUB_1:def 2</a>, <a class="ref" href="prob_1.html#D1" title="PROB_1:def.1">PROB_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
