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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="abcmiz_1.html#K49" title="ABCMIZ_1:func.49">VarPoset</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c1">X</font>, <a href="abcmiz_1.html#K49" title="ABCMIZ_1:func.49">VarPoset</a>  &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">a</font> =  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E1:203"/><span class="lab"><font color="Green" title="E100">A1</font></span>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T3" title="SETFAM_1:th.3">SETFAM_1:3</a></span>;<br/>
<a NAME="E2:203"/><span class="lab"><font color="Green" title="E101">A2</font></span>: 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="abcmiz_1.html#K2" title="ABCMIZ_1:func.2">Vars</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_1.html#T110" target="_self" title="ABCMIZ_1:th.110">Th110</a></span>;<br/>
<a NAME="E3:203"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A3</font></span>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="abcmiz_1.html#K2" title="ABCMIZ_1:func.2">Vars</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:203"><span class="lab"><font color="Green" title="E100">A1</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E4:203"/>
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_1.html#T110" target="_self" title="ABCMIZ_1:th.110">Th110</a></span>;<br/>
<a NAME="E5:203"/><span class="kw">then </span>
 <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_1.html#T12" target="_self" title="ABCMIZ_1:th.12">Th12</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">m</font> =  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c1">X</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="abcmiz_1.html#K49" title="ABCMIZ_1:func.49">VarPoset</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:203"><span class="lab"><font color="Green" title="E100">A1</font></span></a>, <a class="txt" href="#E2:203"><span class="lab"><font color="Green" title="E101">A2</font></span></a>, <a class="txt" href="#E3:203"><span class="lab"><font color="Green" title="E102">A3</font></span></a>, <a class="ref" href="abcmiz_1.html#T110" target="_self" title="ABCMIZ_1:th.110">Th110</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E103">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="203_1"><span class="kw">now </span></a><div class="add"><span class="kw">thus </span><a NAME="E1:203_1"/>
<font color="Maroon" title="c1">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c3">m</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="abcmiz_1.html#K49" title="ABCMIZ_1:func.49">VarPoset</a>  st <font color="Maroon" title="c1">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c3">m</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">b</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="203_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">n</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="abcmiz_1.html#K49" title="ABCMIZ_1:func.49">VarPoset</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c4">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:203_1_1"/>
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font></span><br/>

<a NAME="E2:203_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T3" title="SETFAM_1:th.3">SETFAM_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:203_1_1"/>
<font color="Maroon" title="c4">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_1.html#T109" target="_self" title="ABCMIZ_1:th.109">Th109</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="abcmiz_1.html#K49" title="ABCMIZ_1:func.49">VarPoset</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c3">m</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">b</font> )</span><br/><span class="kw">assume </span><a NAME="E2:203_1"/><span class="lab"><font color="Green" title="E104">A5</font></span>: 
<font color="Maroon" title="c1">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">m</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">b</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="203_1_2"><span class="kw">now </span></a><div class="add"><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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c4">b</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="E1:203_1_2"/><span class="lab"><font color="Green" title="E105">A6</font></span>: 
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">c</font> = <font color="Maroon" title="c5">Y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="abcmiz_1.html#K49" title="ABCMIZ_1:func.49">VarPoset</a> ;<br/><a NAME="E3:203_1_2"/>
<font color="Maroon" title="c6">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:203_1"><span class="lab"><font color="Green" title="E104">A5</font></span></a>, <a class="txt" href="#E1:203_1_2"><span class="lab"><font color="Green" title="E105">A6</font></span></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E4:203_1_2"/>
<font color="Maroon" title="c4">b</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="ref" href="abcmiz_1.html#T109" target="_self" title="ABCMIZ_1:th.109">Th109</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:203_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T5" title="SETFAM_1:th.5">SETFAM_1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E5:203_1"/>
<font color="Maroon" title="c3">m</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_1.html#T109" target="_self" title="ABCMIZ_1:th.109">Th109</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:203"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c1">X</font>, <a href="abcmiz_1.html#K49" title="ABCMIZ_1:func.49">VarPoset</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T15" title="YELLOW_0:th.15">YELLOW_0:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c1">X</font></span><br/>

<span class="kw">hence </span><a NAME="E9:203"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:203"><span class="lab"><font color="Green" title="E103">A4</font></span></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
