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

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/>

<b>assume </b><a NAME="E1:47_1_1"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c2">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>
<b>thus </b><a NAME="E3:47_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_5.html#D3" target="_self" title="WAYBEL_5:def.3">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" title="c3">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="lattice3.html#D12" title="LATTICE3:def.12">LATTICE3:def 12</a><br/>

<b>take </b>
<font color="Maroon" title="c2">x</font>
;<br/>

<b>thus </b><a NAME="E1:47_1_1_1"/>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c2">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" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a><br/>

<b>thus </b><a NAME="E1:47_1_1_1_1"/>
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> )
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:47_1_1"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<a NAME="E2:47_1_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E1:47_1_1"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E3:47_1_1_1"/>
(  not <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c4">y</font> or <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#T24" title="ORDERS_2:th.24">ORDERS_2:24</a></i>;<br/>


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

<a NAME="E4:47_1_1"/><b>then </b>
<font color="Maroon" title="c1">L</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">L'</font> = <font color="Maroon" title="c1">L</font> as  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> ;<br/>
<a NAME="E6:47_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L'</font> holds  <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E1:47_1_1"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>;<br/>
<a NAME="E7:47_1_1"/><b>then </b><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c3">L'</font> is <a href="waybel_3.html#V2" title="WAYBEL_3:attr.2">satisfying_axiom_of_approximation</a>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D5" title="WAYBEL_3:def.5">WAYBEL_3:def 5</a></i>;<br/>
<a NAME="E8:47_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L'</font> holds <br/> ( not  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">L'</font> = <font color="Maroon" title="c3">L'</font> as  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> <b>by </b><i><a class="txt" href="waybel_5.html#E7:47_1_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="waybel_3.html#D6" title="WAYBEL_3:def.6">WAYBEL_3:def 6</a></i>;<br/>
<a NAME="E10:47_1_1"/>
for <font color="Olive" title="b1">J</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">K</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V7</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Olive" title="b1">J</font><br/> for <font color="Olive" title="b3">F</font> being  <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Olive" title="b2">K</font>,<font color="Maroon" title="c4">L'</font> holds   <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">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" title="c5">J</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c6">K</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V7</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c5">J</font>;<br/><b>let </b><font color="Maroon" title="c7">F</font> be   <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c4">L'</font>;<br/>





<a NAME="E1:47_1_1_2"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">J</font> holds   <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">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" title="c8">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">J</font>;<br/>

<a NAME="E1:47_1_1_2_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span> holds <br/> ex <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span> &amp; <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">z</font> &amp; <font color="Olive" title="b2">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">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" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><a NAME="E1:47_1_1_2_1_1"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c11">z</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">z</font> = <font color="Maroon" title="c11">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c12">z</font>
;<br/>

<b>thus </b><a NAME="E4:47_1_1_2_1_1"/>
( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span> &amp; <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">z</font> &amp; <font color="Maroon" title="c10">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">z</font> )
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:47_1_1"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</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" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">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" title="YELLOW_2:NK.7">Inf</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 <b>by </b><i><a class="txt" href="waybel_5.html#E27"><i><font color="Green" title="E22">Lm8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:47_1_1"/>
for <font color="Olive" title="b1">J</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">K</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V7</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Olive" title="b1">J</font><br/> for <font color="Olive" title="b3">F</font> being  <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Olive" title="b2">K</font>,<font color="Maroon" title="c1">L</font> holds   <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 ;<br/>


</div>
