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

<b>let </b><font color="Maroon" title="c1">A</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> ; <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">L</font> being <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a><br/> for <font color="Olive" title="b2">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b3">d</font> being  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">L</font><br/> for <font color="Olive" title="b4">q</font> being   <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Olive" title="b3">d</font> holds <br/> ( <font color="Olive" title="b2">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Olive" title="b3">d</font> iff <font color="Olive" title="b2">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b4">q</font> )</span><br/><b>let </b><font color="Maroon" title="c2">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</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">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b2">d</font> being  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b3">q</font> being   <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Olive" title="b2">d</font> holds <br/> ( <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Olive" title="b2">d</font> iff <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b3">q</font> )</span><br/><b>let </b><font color="Maroon" title="c3">O</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</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">d</font> being  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b2">q</font> being   <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Olive" title="b1">d</font> holds <br/> ( <font color="Maroon" title="c3">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Olive" title="b1">d</font> iff <font color="Maroon" title="c3">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">q</font> )</span><br/><b>let </b><font color="Maroon" title="c4">d</font> be   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">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">q</font> being   <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Maroon" title="c4">d</font> holds <br/> ( <font color="Maroon" title="c3">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c4">d</font> iff <font color="Maroon" title="c3">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">q</font> )</span><br/><b>let </b><font color="Maroon" title="c5">q</font> be    <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Maroon" title="c4">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c4">d</font> iff <font color="Maroon" title="c3">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">q</font> )</span><br/>









<a NAME="E1:53"/>
 <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c4">d</font>,<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>, <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> : <font color="Maroon" title="c4">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">x</font>,<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">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b4">b</font> </span>}</span>  <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="lattice5.html#D12" target="_self" title="LATTICE5:def.12">Def12</a></i>;<br/>
<a NAME="E2:53"/><b>then </b><i><font color="Green" title="E38">A1</font></i>: 
 <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c4">d</font>, <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="lattice5.html#D14" target="_self" title="LATTICE5:def.14">Def14</a></i>;<br/>
<a NAME="E3:53"/>
<font color="Maroon" title="c5">q</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="lattice5.html#D14" target="_self" title="LATTICE5:def.14">Def14</a></i>;<br/>
<a NAME="E4:53"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">q</font>, <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E5:53"/><b>then </b><i><font color="Green" title="E39">A2</font></i>: 
 <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c4">d</font>, <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E2:53"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="wellord2.html#T22" title="WELLORD2:th.22">WELLORD2:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">M</font> =  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c4">d</font> as   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">N</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">q</font> as   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> <b>by </b><i><a class="ref" href="lattice5.html#D14" target="_self" title="LATTICE5:def.14">Def14</a></i>;<br/>
<a NAME="E8:53"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c6">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">N</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E5:53"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="card_1.html#T8" title="CARD_1:th.8">CARD_1:8</a></i>;<br/>
<b>hence </b><a NAME="E9:53"/>
( <font color="Maroon" title="c3">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c4">d</font> implies <font color="Maroon" title="c3">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">q</font> implies <font color="Maroon" title="c3">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c4">d</font> )</span><br/>

<b>assume </b><a NAME="E10:53"/>
<font color="Maroon" title="c3">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c4">d</font></span><br/>

<b>hence </b><a NAME="E11:53"/>
<font color="Maroon" title="c3">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E8:53"><i><font color="Green" title="E40">A3</font></i></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>
