<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">Omega</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font>;<br/>



<b>set </b><font color="Maroon">V</font> = <span class="p1">{<span class="default"> <font color="Olive">M</font> where <font color="Olive">M</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">Y</font> =  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">M</font> where <font color="Olive">M</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> ) </span>}</span> ;<br/>
<a NAME="E1:77"/><i><font color="Green">E36</font></i>: 
 <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">Omega</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">M</font> where <font color="Olive">M</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>proof </b></a><div class="add">

<a NAME="E1:77_1"/>
 <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">Omega</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font>
 
<b>by </b><i><a class="ref" href="prob_3.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E2:77_1"/>
 <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">Omega</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">M</font> where <font color="Olive">M</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:77"/><b>then </b><i><font color="Green">E52</font></i>: 
(  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">M</font> where <font color="Olive">M</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">Omega</font> &amp; <span class="p1">{<span class="default"> <font color="Olive">M</font> where <font color="Olive">M</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<a NAME="E3:77"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">M</font> where <font color="Olive">M</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> ) </span>}</span>  holds <br/><font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:77_2"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">M</font> where <font color="Olive">M</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:77_2"/><b>then </b>
 ex <font color="Olive">M</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> st <br/>( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> )
 
;<br/>
<b>hence </b><a NAME="E3:77_2"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:77"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">MSeq</font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">Omega</font>  st <font color="Olive">MSeq</font> is <a href="setlim_1.html#V1">monotone</a> &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Olive">MSeq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">M</font> where <font color="Olive">M</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> ) </span>}</span>  ) holds <br/> <a href="setlim_1.html#K3">lim</a> <font color="Olive">MSeq</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">M</font> where <font color="Olive">M</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">MSeq</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">Omega</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:77_3"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">MSeq</font> is <a href="setlim_1.html#V1">monotone</a>
 <b>and </b><br/><a NAME="E2:77_3"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">MSeq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">M</font> where <font color="Olive">M</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:77_3_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">M</font> where <font color="Olive">M</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 ;<br/><a NAME="E2:77_3_1"/><b>then </b><i><font color="Green">E76</font></i>: 
 ex <font color="Olive">M</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> st <br/>( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:77_3_1_1"/>
<font color="Maroon">MSeq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">M</font> where <font color="Olive">M</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="prob_3.html#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E2:77_3_1_1"/>
<font color="Maroon">MSeq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="prob_3.html#T13" target="_self">Th13</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:77_3_1"/>
 <a href="setlim_1.html#K3">lim</a> <font color="Maroon">MSeq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="prob_3.html#T11" target="_self">Th11</a>, <a class="ref" href="prob_3.html#T14" target="_self">Th14</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:77_3"/>
 <a href="setlim_1.html#K3">lim</a> <font color="Maroon">MSeq</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">M</font> where <font color="Olive">M</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">Y</font> =  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">M</font> where <font color="Olive">M</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> ) </span>}</span>  as    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> <b>by </b><i>, <a class="ref" href="prob_3.html#T10" target="_self">Th10</a>, </i>;<br/>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<a NAME="E6:77"/>
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font> &amp; <font color="Olive">Z</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> holds <br/><font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:77_4"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> )
 ;<br/>

<a NAME="E2:77_4"/><b>then </b>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">M</font> where <font color="Olive">M</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E3:77_4"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:77"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font> &amp; <font color="Olive">Z</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">Omega</font> holds <br/><font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font> ) )
 <b>by </b><i><a class="ref" href="prob_3.html#D2" target="_self">Def2</a>, , <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>


</div>
