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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c4">X'</font>, <font color="Maroon" title="c5">Y'</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:5"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X'</font>
 <b>and </b><br/><a NAME="E2:5"/><i><font color="Green" title="E5">A2</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Y'</font>
 ;<br/>

<b>thus </b><a NAME="E3:5"/>
<font color="Maroon" title="c2">X</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X'</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c5">Y'</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">Y</font>
 ;<br/>

<a NAME="E2:5_1"/><b>then </b>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">q</font></span>)</span> where <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#D3" title="YELLOW_4:def.3">YELLOW_4:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E4:5_1"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c8">y</font>
 <b>and </b><br/><a NAME="E5:5_1"/><i><font color="Green" title="E7">A4</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E6:5_1"/><i><font color="Green" title="E8">A5</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ;<br/>
<a NAME="E7:5_1"/><i><font color="Green" title="E9">A6</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X'</font> &amp; <font color="Maroon" title="c8">y</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y'</font> )
 
<b>by </b><i><a class="txt" href="waybel16.html#E1:5"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="waybel16.html#E2:5"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="waybel16.html#E5:5_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="waybel16.html#E6:5_1"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="lattice3.html#D6" title="LATTICE3:def.6">LATTICE3:def 6</a></i>;<br/>
<a NAME="E8:5_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="waybel16.html#E4:5_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="yellow_7.html#T23" title="YELLOW_7:th.23">YELLOW_7:23</a></i>;<br/>
<a NAME="E9:5_1"/><b>then </b>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b2">q</font></span>)</span> where <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X'</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y'</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel16.html#E7:5_1"><i><font color="Green" title="E9">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:5_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X'</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c5">Y'</font>
 <b>by </b><i><a class="ref" href="yellow_4.html#D4" title="YELLOW_4:def.4">YELLOW_4:def 4</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c6">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:5"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X'</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c5">Y'</font>
 ;<br/>

<a NAME="E5:5"/><b>then </b>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b2">q</font></span>)</span> where <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X'</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y'</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#D4" title="YELLOW_4:def.4">YELLOW_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span><b> such that </b><br/><a NAME="E7:5"/><i><font color="Green" title="E6">A7</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c8">y</font>
 <b>and </b><br/><a NAME="E8:5"/><i><font color="Green" title="E7">A8</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X'</font>
 <b>and </b><br/><a NAME="E9:5"/><i><font color="Green" title="E8">A9</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y'</font>
 ;<br/>
<a NAME="E10:5"/><i><font color="Green" title="E9">A10</font></i>: 
(  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> )
 
<b>by </b><i><a class="txt" href="waybel16.html#E1:5"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="waybel16.html#E2:5"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="waybel16.html#E8:5"><i><font color="Green" title="E7">A8</font></i></a>, <a class="txt" href="waybel16.html#E9:5"><i><font color="Green" title="E8">A9</font></i></a>, <a class="ref" href="lattice3.html#D7" title="LATTICE3:def.7">LATTICE3:def 7</a></i>;<br/>
<a NAME="E11:5"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c8">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel16.html#E7:5"><i><font color="Green" title="E6">A7</font></i></a>, <a class="ref" href="yellow_7.html#T24" title="YELLOW_7:th.24">YELLOW_7:24</a></i>;<br/>
<a NAME="E12:5"/><b>then </b>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">q</font></span>)</span> where <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel16.html#E10:5"><i><font color="Green" title="E9">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:5"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="ref" href="yellow_4.html#D3" title="YELLOW_4:def.3">YELLOW_4:def 3</a></i>;<br/>


</div>
