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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>consider </b><font color="Maroon" title="c2">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E2:203"/>
(  <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> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c2">a</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>  )
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T110" target="_self" title="ABCMIZ_1:th.110">Th21</a>, <a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></i>;<br/>
<a NAME="E3:203"/><b>then </b><i><font color="Green" title="E100">A1</font></i>: 
(  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c1">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <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>  )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<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>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T110" target="_self" title="ABCMIZ_1:th.110">Th21</a></i>;<br/>
<a NAME="E5:203"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T12" target="_self" title="ABCMIZ_1:th.12">Th8</a></i>;<br/>
<b>then reconsider </b><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>  <b>by </b><i><a class="txt" href="#E3:203"><i><font color="Green" title="E100">A1</font></i></a>, <a class="ref" href="abcmiz_1.html#T110" target="_self" title="ABCMIZ_1:th.110">Th21</a></i>;<br/>
<div><i><font color="Green" title="E101">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="203_1"><b>now </b></a><div class="add"><b>thus </b><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:()"><i><font color="Red">::  thesis: </font></i></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"><b>proof </b></a><div class="add">

<b>let </b><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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="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/>

<b>assume </b><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:()"><i><font color="Red">::  thesis: </font></i></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"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></i>;<br/>
<b>hence </b><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>
 <b>by </b><i><a class="ref" href="abcmiz_1.html#T109" target="_self" title="ABCMIZ_1:th.109">Th22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>assume </b><a NAME="E2:203_1"/><i><font color="Green" title="E102">A2</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">Y</font> <a href="hidden.html#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/><b>assume </b><a NAME="E1:203_1_2"/><i><font color="Green" title="E103">A3</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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/><b>then reconsider </b><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>
 <b>by </b><i><a class="txt" href="#E2:203_1"><i><font color="Green" title="E102">A2</font></i></a>, <a class="txt" href="#E1:203_1_2"><i><font color="Green" title="E103">A3</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/><b>hence </b><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>
 <b>by </b><i><a class="ref" href="abcmiz_1.html#T109" target="_self" title="ABCMIZ_1:th.109">Th22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E4:203_1"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="setfam_1.html#T6" title="SETFAM_1:th.6">SETFAM_1:6</a></i>;<br/><b>hence </b><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>
 <b>by </b><i><a class="ref" href="abcmiz_1.html#T109" target="_self" title="ABCMIZ_1:th.109">Th22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="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> 
 <b>by </b><i><a class="ref" href="yellow_0.html#T15" title="YELLOW_0:th.15">YELLOW_0:15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>hence </b><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>
 <b>by </b><i><a class="txt" href="#E7:203"><i><font color="Green" title="E101">A4</font></i></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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