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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c4">x'</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c5">y'</font> = <font color="Maroon" title="c3">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/><a NAME="E2:8_1"/>
( <font color="Maroon" title="c4">x'</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> &amp; <font color="Maroon" title="c5">y'</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> )
 <b>by </b><i><a class="ref" href="waybel_8.html#D1" title="WAYBEL_8:def.1">WAYBEL_8:def 1</a></i>;<br/><a NAME="E3:8_1"/><b>then </b>
( <font color="Maroon" title="c4">x'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c5">y'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="ref" href="waybel_8.html#T30" title="WAYBEL_8:th.30">WAYBEL_8:30</a></i>;<br/><a NAME="E4:8_1"/><b>then </b>
<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> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T15" title="FINSET_1:th.15">FINSET_1:15</a></i>;<br/><a NAME="E5:8_1"/><b>then </b>
<font color="Maroon" title="c4">x'</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="yellow_1.html#T17" title="YELLOW_1:th.17">YELLOW_1:17</a></i>;<br/><a NAME="E6:8_1"/><b>then </b>
<font color="Maroon" title="c4">x'</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y'</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 <b>by </b><i><a class="ref" href="waybel_8.html#T30" title="WAYBEL_8:th.30">WAYBEL_8:30</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">xy</font> = <font color="Maroon" title="c4">x'</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="waybel_8.html#D1" title="WAYBEL_8:def.1">WAYBEL_8:def 1</a></i>;<br/><a NAME="E8:8_1"/>
( <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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x'</font> &amp; <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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">y'</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E9:8_1"/><b>then </b>
( <font color="Maroon" title="c4">x'</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x'</font> &amp; <font color="Maroon" title="c4">x'</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y'</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="yellow_1.html#T17" title="YELLOW_1:th.17">YELLOW_1:17</a></i>;<br/><a NAME="E10:8_1"/><b>then </b>
( <font color="Maroon" title="c4">x'</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x'</font> &amp; <font color="Maroon" title="c4">x'</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y'</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T2" title="YELLOW_1:th.2">YELLOW_1:2</a></i>;<br/><a NAME="E11:8_1"/><b>then </b><i><font color="Green" title="E7">A1</font></i>: 
( <font color="Maroon" title="c6">xy</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c6">xy</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c8">z'</font> = <font color="Maroon" title="c7">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/><b>assume </b><a NAME="E2:8_1_1"/>
( <font color="Maroon" title="c7">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c7">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> )
 ;<br/><a NAME="E3:8_1_1"/><b>then </b>
( <font color="Maroon" title="c8">z'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x'</font> &amp; <font color="Maroon" title="c8">z'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y'</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T60" title="YELLOW_0:th.60">YELLOW_0:60</a></i>;<br/><a NAME="E4:8_1_1"/><b>then </b>
( <font color="Maroon" title="c8">z'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x'</font> &amp; <font color="Maroon" title="c8">z'</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="yellow_1.html#T2" title="YELLOW_1:th.2">YELLOW_1:2</a></i>;<br/><a NAME="E5:8_1_1"/><b>then </b>
<font color="Maroon" title="c8">z'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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>
 <b>by </b><i><a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/><a NAME="E6:8_1_1"/><b>then </b>
<font color="Maroon" title="c8">z'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x'</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y'</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T17" title="YELLOW_1:th.17">YELLOW_1:17</a></i>;<br/><a NAME="E7:8_1_1"/><b>then </b>
<font color="Maroon" title="c8">z'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x'</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y'</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T2" title="YELLOW_1:th.2">YELLOW_1:2</a></i>;<br/><b>hence </b><a NAME="E8:8_1_1"/>
<font color="Maroon" title="c6">xy</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">z</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:8_1"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel15.html#E11:8_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="yellow_0.html#T19" title="YELLOW_0:th.19">YELLOW_0:19</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:8"/><b>then </b>
 <a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span> is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T21" title="YELLOW_0:th.21">YELLOW_0:21</a></i>;<br/>
<b>hence </b><a NAME="E3:8"/>
 <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c1">X</font> is <a href="waybel_8.html#V3" title="WAYBEL_8:attr.3">arithmetic</a>
 <b>by </b><i><a class="ref" href="waybel_8.html#T21" title="WAYBEL_8:th.21">WAYBEL_8:21</a></i>;<br/>


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