<?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> ;<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>;<br/><b>let </b><font color="Maroon" title="c3">d</font> be   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</font>;<br/><b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D34" title="WAYBEL_0:def.34">WAYBEL_0:def 34</a><br/>







<b>set </b><font color="Maroon" title="c6">f</font> =  <a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c3">d</font>;<br/>
<a NAME="E1:34"/><i><font color="Green" title="E22">A1</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>
<a NAME="E2:34"/><i><font color="Green" title="E23">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">E1</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E4:34"/><i><font color="Green" title="E24">A3</font></i>: 
( <font color="Maroon" title="c7">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E1</font> iff <font color="Maroon" title="c3">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="Maroon" title="c4">a</font> ) ) )
 <b>by </b><i><a class="ref" href="lattice5.html#D9" target="_self" title="LATTICE5:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">E2</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E6:34"/><i><font color="Green" title="E25">A4</font></i>: 
( <font color="Maroon" title="c8">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E2</font> iff <font color="Maroon" title="c3">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="Maroon" title="c5">b</font> ) ) )
 <b>by </b><i><a class="ref" href="lattice5.html#D9" target="_self" title="LATTICE5:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">E3</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E8:34"/><i><font color="Green" title="E26">A5</font></i>: 
( <font color="Maroon" title="c9">E3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">b</font></span>)</span> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">E3</font> iff <font color="Maroon" title="c3">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="Maroon" title="c4">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">b</font> ) ) )
 <b>by </b><i><a class="ref" href="lattice5.html#D9" target="_self" title="LATTICE5:def.9">Def9</a></i>;<br/>
<a NAME="E9:34"/><i><font color="Green" title="E27">A6</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E1</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c8">E2</font> iff <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">E3</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:34_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E1</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c8">E2</font>
 ;<br/><a NAME="E2:34_1_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:34_1_1"/><b>then </b>
( <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E4:34"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="lattice5.html#E6:34"><i><font color="Green" title="E25">A4</font></i></a></i>;<br/><a NAME="E4:34_1_1"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><b>hence </b><a NAME="E5:34_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">E3</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E8:34"><i><font color="Green" title="E26">A5</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:34_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">E3</font>
 ;<br/>

<a NAME="E3:34_1"/><b>then </b><i><font color="Green" title="E28">A7</font></i>: 
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E8:34"><i><font color="Green" title="E26">A5</font></i></a></i>;<br/>
<a NAME="E4:34_1"/>
<font color="Maroon" title="c4">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">a</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/>
<a NAME="E5:34_1"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">a</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:34_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>
<a NAME="E6:34_1"/><b>then </b><i><font color="Green" title="E29">A8</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E1</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E4:34"><i><font color="Green" title="E24">A3</font></i></a></i>;<br/>
<a NAME="E7:34_1"/>
<font color="Maroon" title="c4">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/>
<a NAME="E8:34_1"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:34_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>
<a NAME="E9:34_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E2</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E6:34"><i><font color="Green" title="E25">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:34_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E1</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c8">E2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E6:34_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:34"/><i><font color="Green" title="E28">A9</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E1</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c8">E2</font> iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">E3</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E1:34_2"/><i><font color="Green" title="E29">A10</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">E3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T16" title="EQREL_1:th.16">EQREL_1:16</a></i>;<br/>
<span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c7">E1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c8">E2</font></span>)</span> = 
<font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c8">E2</font></span>)</span>
<b>by </b><i><a class="ref" href="eqrel_1.html#T16" title="EQREL_1:th.16">EQREL_1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">A</font>
<b>by </b><i><a class="ref" href="eqrel_1.html#T16" title="EQREL_1:th.16">EQREL_1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">A</font>

;<br/>
<a NAME="E3:34_2"/><b>then </b><i><font color="Green" title="E30">A11</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E1</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c8">E2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T34" title="RELAT_1:th.34">RELAT_1:34</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:34_2_2"/><i><font color="Green" title="E31">A12</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E1</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c8">E2</font>
 ;<br/><a NAME="E2:34_2_2"/><b>then </b>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E1</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c8">E2</font></span>)</span> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E1</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c8">E2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">x'</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c13">y'</font> = <font color="Maroon" title="c11">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="lattice5.html#E3:34_2"><i><font color="Green" title="E30">A11</font></i></a></i>;<br/><a NAME="E4:34_2_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x'</font>,<font color="Maroon" title="c13">y'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">E3</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E9:34"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="lattice5.html#E1:34_2_2"><i><font color="Green" title="E31">A12</font></i></a></i>;<br/><b>hence </b><a NAME="E5:34_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">E3</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:34_2"/><i><font color="Green" title="E31">A13</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">E3</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c12">x'</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c13">y'</font> = <font color="Maroon" title="c11">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="lattice5.html#E1:34_2"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<a NAME="E7:34_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x'</font>,<font color="Maroon" title="c13">y'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E1</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c8">E2</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E9:34"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="lattice5.html#E5:34_2"><i><font color="Green" title="E31">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:34_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E1</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c8">E2</font>
 ;<br/>


</div><b>end;</b></div>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> = 
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E2:34"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="funct_1.html#T118" title="FUNCT_1:th.118">FUNCT_1:118</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#T40" title="YELLOW_0:th.40">YELLOW_0:40</a></i>
<br/>.= 
<font color="Maroon" title="c7">E1</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c8">E2</font>
<b>by </b><i><a class="txt" href="lattice5.html#E4:34"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="lattice5.html#E6:34"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="lattice5.html#T8" target="_self" title="LATTICE5:th.8">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E8:34"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="lattice5.html#E10:34"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="relat_1.html#D2" title="RELAT_1:def.2">RELAT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#T40" title="YELLOW_0:th.40">YELLOW_0:40</a></i>
;<br/>
<b>hence </b><a NAME="E12:34"/>
 <a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c3">d</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:34"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="waybel_0.html#D30" title="WAYBEL_0:def.30">WAYBEL_0:def 30</a></i>;<br/>


</div>
