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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span>;<br/>



<a NAME="E1:50"/>
 <a href="lattice3.html#K3">LattPOSet</a> <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> <font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice3.html#K2">LattRel</a> <span class="p2">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> <font color="Maroon">X</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="lattice3.html#D2">LATTICE3:def 2</a></i>;<br/>
<b>then </b><i><font color="Green">E31</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#D2">YELLOW_1:def 2</a></i>
<br/>.= 
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="lattice3.html#D1">LATTICE3:def 1</a></i>
;<br/>
<b>thus </b><a NAME="E3:50"/>
( <font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font> implies for <font color="Olive">Y</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>  st <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">Y</font> holds <br/> ex <font color="Olive">Z</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Olive">Y</font> st <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Olive">Z</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 ;<br/>

<b>let </b><font color="Maroon">Y</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>

<b>reconsider </b><font color="Maroon">Y'</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E3:50_2"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E4:50_2"/><b>then </b>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y'</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T21">YELLOW_1:21</a></i>;<br/>
<a NAME="E5:50_2"/><b>then </b>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y'</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/>
<b>then consider </b><font color="Maroon">Z</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E7:50_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T18">WAYBEL_3:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Z'</font> = <font color="Maroon">Z</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="waybel_8.html#D1" target="_self">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon">Z'</font>
;<br/>

<a NAME="E9:50_2"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="waybel_8.html#D1" target="_self">Def1</a>, <a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/>
<b>hence </b><a NAME="E10:50_2"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z'</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T21">YELLOW_1:21</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:50"/>
( ( for <font color="Olive">Y</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>  st <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">Y</font> holds <br/> ex <font color="Olive">Z</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Olive">Y</font> st <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Olive">Z</font> ) implies <font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_3"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">Y</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>  st <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">Y</font> holds <br/> ex <font color="Olive">Z</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Olive">Y</font> st <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Olive">Z</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">Y'</font> = <font color="Maroon">Y</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><b>assume </b><a NAME="E2:50_3_1"/>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E3:50_3_1"/><b>then </b>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/><a NAME="E4:50_3_1"/><b>then </b>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Y'</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T21">YELLOW_1:21</a></i>;<br/><b>then consider </b><font color="Maroon">Z'</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y'</font><b> such that </b><br/><a NAME="E6:50_3_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z'</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Z'</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>take </b><font color="Maroon">Z</font> = <font color="Maroon">Z</font>;<br/><b>thus </b><a NAME="E8:50_3_1"/>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E9:50_3_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="waybel_8.html#D1" target="_self">Def1</a>, <a class="ref" href="yellow_1.html#T21">YELLOW_1:21</a></i>;<br/><b>hence </b><a NAME="E10:50_3_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:50_3"/>
<font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T19">WAYBEL_3:19</a></i>;<br/>


</div><b>end;</b></div>


</div>
