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<b>set </b><font color="Maroon" title="c1">V</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">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>  : verum </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c2">A0</font> being  <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> ;<br/>
<a NAME="E2:201_1_1"/>
 <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon" title="c2">A0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">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>  : verum </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">V</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">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>  : verum </span>}</span>  as  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>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="201_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> implies <font color="Maroon" title="c4">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> )</span><br/><b>assume </b><a NAME="E1:201_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font>
 ; <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="c3">V</font> implies <font color="Maroon" title="c4">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> )</span><br/><b>then consider </b><font color="Maroon" title="c6">A1</font> being  <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> such that </b><br/><a NAME="E3:201_1_1_1"/><i><font color="Green" title="E98">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon" title="c6">A1</font>
 ;<br/><b>assume </b><a NAME="E4:201_1_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</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">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font></span><br/><b>then consider </b><font color="Maroon" title="c7">A2</font> being  <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> such that </b><br/><a NAME="E6:201_1_1_1"/><i><font color="Green" title="E99">A3</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon" title="c7">A2</font>
 ;<br/><a NAME="E7:201_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font></span>)</span>
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E3:201_1_1_1"><i><font color="Green" title="E98">A2</font></i></a>, <a class="txt" href="abcmiz_1.html#E6:201_1_1_1"><i><font color="Green" title="E99">A3</font></i></a>, <a class="ref" href="abcmiz_1.html#T11" target="_self" title="ABCMIZ_1:th.11">Th10</a></i>;<br/><b>hence </b><a NAME="E8:201_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E5:201_1_1"/><b>then </b>
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c3">V</font> is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T11" title="YELLOW_1:th.11">YELLOW_1:11</a></i>;<br/>
<b>hence </b><a NAME="E6:201_1_1"/>
 <a href="abcmiz_1.html#K49" title="ABCMIZ_1:func.49">VarPoset</a>  is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>
 <b>by </b><i><a class="ref" href="lattice3.html#T10" title="LATTICE3:th.10">LATTICE3:10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="abcmiz_1.html#K49" title="ABCMIZ_1:func.49">VarPoset</a>  is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="201_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> implies <font color="Maroon" title="c4">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> )</span><br/><b>assume </b><a NAME="E1:201_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font>
 ; <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="c3">V</font> implies <font color="Maroon" title="c4">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> )</span><br/><b>then consider </b><font color="Maroon" title="c6">A1</font> being  <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> such that </b><br/><a NAME="E3:201_1_1_2"/><i><font color="Green" title="E98">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon" title="c6">A1</font>
 ;<br/><b>assume </b><a NAME="E4:201_1_1_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</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">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font></span><br/><b>then consider </b><font color="Maroon" title="c7">A2</font> being  <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> such that </b><br/><a NAME="E6:201_1_1_2"/><i><font color="Green" title="E99">A3</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon" title="c7">A2</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c8">V1</font> =  <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon" title="c6">A1</font>, <font color="Maroon" title="c9">V2</font> =  <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <font color="Maroon" title="c7">A2</font> as  <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#T24" target="_self" title="ABCMIZ_1:th.24">Th18</a></i>;<br/><a NAME="E8:201_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K1" title="ABCMIZ_1:func.1">varcl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">V2</font></span>)</span>
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E3:201_1_1_2"><i><font color="Green" title="E98">A2</font></i></a>, <a class="txt" href="abcmiz_1.html#E6:201_1_1_2"><i><font color="Green" title="E99">A3</font></i></a>, <a class="ref" href="abcmiz_1.html#T13" target="_self" title="ABCMIZ_1:th.13">Th9</a></i>;<br/><b>hence </b><a NAME="E9:201_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E8:201_1_1"/><b>then </b>
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c3">V</font> is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T12" title="YELLOW_1:th.12">YELLOW_1:12</a></i>;<br/>
<b>hence </b><a NAME="E9:201_1_1"/>
 <a href="abcmiz_1.html#K49" title="ABCMIZ_1:func.49">VarPoset</a>  is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>
 <b>by </b><i><a class="ref" href="yellow_7.html#T16" title="YELLOW_7:th.16">YELLOW_7:16</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>
