<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</font> be  <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> <a href="waybel_0.html#V25" title="WAYBEL_0:attr.25">/\-complete</a> <a href="waybel_0.html#NM6" title="WAYBEL_0:NM.6">Semilattice</a>;<br/>

<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E7">A1</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">L2</font> #)
 ;<br/>

<b>let </b><font color="Maroon" title="c3">N1</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L1</font>;<br/><b>let </b><font color="Maroon" title="c4">N2</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c2">L2</font>;<br/>



<b>assume </b><a NAME="E2:11"/><i><font color="Green" title="E8">A2</font></i>: 
(  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">N1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">N1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">N2</font> #) &amp; the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c4">N2</font> )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N1</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">N1</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N1</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> $1 </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N2</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">N2</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N2</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> $1 </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c5">U1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>),<font color="Maroon" title="c1">L1</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N1</font> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c6">U2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">j</font>),<font color="Maroon" title="c2">L2</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N2</font> : verum </span>}</span> ;<br/>
<a NAME="E3:11"/><i><font color="Green" title="E9">A3</font></i>: 
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c3">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>),<font color="Maroon" title="c1">L1</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N1</font> : verum </span>}</span> ,<font color="Maroon" title="c1">L1</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D6" title="WAYBEL11:def.6">WAYBEL11:def 6</a></i>;<br/>
<a NAME="E4:11"/><i><font color="Green" title="E10">A4</font></i>: 
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c4">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">j</font>),<font color="Maroon" title="c2">L2</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N2</font> : verum </span>}</span> ,<font color="Maroon" title="c2">L2</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D6" title="WAYBEL11:def.6">WAYBEL11:def 6</a></i>;<br/>
<a NAME="E5:11"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>),<font color="Maroon" title="c1">L1</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N1</font> : verum </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</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="E1:11_1"/>
<font color="Maroon" title="c7">x</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="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>),<font color="Maroon" title="c1">L1</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N1</font> : verum </span>}</span> 
 ;<br/>

<a NAME="E2:11_1"/><b>then </b>
 ex <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N1</font> st <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>),<font color="Maroon" title="c1">L1</font>
 
;<br/>
<b>hence </b><a NAME="E3:11_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">U1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>),<font color="Maroon" title="c1">L1</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N1</font> : verum </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L1</font> ;<br/>
<a NAME="E7:11"/>
( not <font color="Maroon" title="c7">U1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c7">U1</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 
<b>by </b><i><a class="ref" href="waybel32.html#T11" title="WAYBEL32:th.11">WAYBEL32:11</a></i>;<br/>
<a NAME="E8:11"/><b>then </b><i><font color="Green" title="E11">A5</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c7">U1</font>,<font color="Maroon" title="c1">L1</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75" title="WAYBEL_0:th.75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E9:11"/>
<font color="Maroon" title="c7">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">j</font>),<font color="Maroon" title="c2">L2</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N2</font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:11_2"/>
( <font color="Maroon" title="c7">U1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">j</font>),<font color="Maroon" title="c2">L2</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N2</font> : verum </span>}</span>  &amp; <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">j</font>),<font color="Maroon" title="c2">L2</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N2</font> : verum </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">U1</font> )
 <b>by </b><i><a class="txt" href="waybel33.html#E1:11"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="waybel33.html#E2:11"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="waybel33.html#E7"><i><font color="Green" title="E5">Lm2</font></i></a></i>; <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><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:11"/>
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c3">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c4">N2</font>
 <b>by </b><i><a class="txt" href="waybel33.html#E1:11"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="waybel33.html#E3:11"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="waybel33.html#E4:11"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="waybel33.html#E8:11"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="yellow_0.html#T26" title="YELLOW_0:th.26">YELLOW_0:26</a></i>;<br/>


</div>
