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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font>  st ( for <font color="Olive" title="b2">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/>( <font color="Olive" title="b2">X</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> &amp; <font color="Olive" title="b2">X</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> ) ) holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">F</font> is  <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font></span><br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> <font color="Maroon" title="c2">F</font> holds <br/>( <font color="Olive" title="b1">X</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> &amp; <font color="Olive" title="b1">X</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> ) ) implies  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font> is  <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> )</span><br/>



<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E20">A1</font></i>: 
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> <font color="Maroon" title="c2">F</font> holds <br/>( <font color="Olive" title="b1">X</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> &amp; <font color="Olive" title="b1">X</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font> is  <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c3">F'</font> = <font color="Maroon" title="c2">F</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">M</font> =  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c3">F'</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1"/>
( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1"><b>suppose </b></a><a NAME="E1:45_1_1"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font> is  <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font></span><br/><div class="add"><a NAME="E2:45_1_1"/>
<font color="Maroon" title="c4">M</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">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="waybel_0.html#D2" title="WAYBEL_0:def.2">WAYBEL_0:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> or  not <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> ) )</span><br/>

<b>assume </b><a NAME="E1:45_1_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> )</span><br/>

<b>hence </b><a NAME="E2:45_1_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="txt" href="#E1:45_1_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:45_1_1"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font> is  <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2"><b>suppose </b></a><a NAME="E1:45_1_2"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font> is  <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font></span><br/><div class="add"><a NAME="E2:45_1_2"/>
<font color="Maroon" title="c4">M</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">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="waybel_0.html#D2" title="WAYBEL_0:def.2">WAYBEL_0:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> or  not <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> ) )</span><br/>

<b>assume </b><a NAME="E1:45_1_2_1"/><i><font color="Green" title="E22">A4</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> )</span><br/>

<b>take </b><font color="Maroon" title="c7">z</font> = <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c7">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c7">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> )</span><br/>

<a NAME="E2:45_1_2_1"/>
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> implies <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> )</span><br/>

<b>assume </b><a NAME="E1:45_1_2_1_1"/><i><font color="Green" title="E23">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="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c9">X</font> = <font color="Maroon" title="c8">Y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E3:45_1_2_1_1"/><i><font color="Green" title="E24">A6</font></i>: 
( <font color="Maroon" title="c9">X</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> &amp; <font color="Maroon" title="c9">X</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> )
 
<b>by </b><i><a class="txt" href="#E1:45"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1"><i><font color="Green" title="E23">A5</font></i></a></i>;<br/>
<a NAME="E4:45_1_2_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X</font> )
 
<b>by </b><i><a class="txt" href="#E1:45_1_2_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:45_1_2_1_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="waybel_0.html#T41" title="WAYBEL_0:th.41">WAYBEL_0:41</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:45_1_2_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="txt" href="#E1:45_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c7">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> )</span><br/>

<b>thus </b><a NAME="E4:45_1_2_1"/>
( <font color="Maroon" title="c7">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c7">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:45_1_2"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font> is  <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
