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

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>assume </b><a NAME="E1:47_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">L</font> is <a href="realset2.html#V3">trivial</a>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>
<b>thus </b><a NAME="E3:47_1_1"/>
<font color="Maroon">L</font> is <a href="lattice3.html#V3">complete</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_5.html#D3" target="_self">WAYBEL_5:def 3</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D12">LATTICE3:def 12</a><br/>

<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E1:47_1_1_1"/>
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>thus </b><a NAME="E1:47_1_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> implies <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>


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

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<a NAME="E2:47_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E3:47_1_1_1"/>
(  not <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">y</font> or <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#T24">ORDERS_2:24</a></i>;<br/>


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

<a NAME="E4:47_1_1"/><b>then </b>
<font color="Maroon">L</font> is  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="orders_2.html#NM1">Poset</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon">L'</font> = <font color="Maroon">L</font> as  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a> ;<br/>
<a NAME="E6:47_1_1"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> holds  <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<a NAME="E7:47_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">L'</font> is <a href="waybel_3.html#V2">satisfying_axiom_of_approximation</a>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a></i>;<br/>
<a NAME="E8:47_1_1"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> holds <br/> ( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is <a href="waybel_0.html#V1">directed</a> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">L'</font> = <font color="Maroon">L'</font> as  <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i><a class="txt" href="waybel_5.html#E1:47_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a></i>;<br/>
<a NAME="E10:47_1_1"/>
for <font color="Olive">J</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">K</font> being <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">J</font><br/> for <font color="Olive">F</font> being  <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Olive">K</font>,<font color="Maroon">L'</font> holds   <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">J</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">K</font> be  <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">K</font>,<font color="Maroon">L'</font>;<br/>





<a NAME="E1:47_1_1_2"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> holds   <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">j</font></span>)</span> is <a href="waybel_0.html#V1">directed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font>;<br/>

<a NAME="E1:47_1_1_2_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> holds <br/> ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font> &amp; <font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:47_1_1_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> )
 ;<br/>

<b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<b>thus </b><a NAME="E4:47_1_1_2_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i>, <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:47_1_1_2_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> is <a href="waybel_0.html#V1">directed</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:47_1_1_2"/>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:47_1_1"/>
for <font color="Olive">J</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">K</font> being <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">J</font><br/> for <font color="Olive">F</font> being  <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Olive">K</font>,<font color="Maroon">L</font> holds   <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 ;<br/>


</div>
