<?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>assume </b><a NAME="E1:35"/>
<font color="Maroon" title="c3">d</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 ;<br/>

<a NAME="E2:35"/><b>then </b><i><font color="Green" title="E23">A1</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">d</font> <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#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">f</font> =  <a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c3">d</font>;<br/>
<a NAME="E3:35"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>  st <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="Olive" title="b1">a</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="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>;<br/>

<b>assume </b><a NAME="E1:35_1"/><i><font color="Green" title="E24">A2</font></i>: 
<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">a</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="c6">b</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:35_1"/><i><font color="Green" title="E25">A3</font></i>: 
( <font color="Maroon" title="c7">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c3">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E2:35"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:35_1"/><i><font color="Green" title="E26">A4</font></i>: 
( <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c7">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="lattice5.html#E3:35_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:35_1"/><i><font color="Green" title="E27">A5</font></i>: 
( <font color="Maroon" title="c10">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c3">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E2:35"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">x2</font>, <font color="Maroon" title="c12">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:35_1"/><i><font color="Green" title="E28">A6</font></i>: 
( <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c10">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x2</font>,<font color="Maroon" title="c12">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="lattice5.html#E7:35_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">x1</font> = <font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c14">y1</font> = <font color="Maroon" title="c9">y1</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#E5:35_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x2</font> = <font color="Maroon" title="c11">x2</font>, <font color="Maroon" title="c16">y2</font> = <font color="Maroon" title="c12">y2</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#E9:35_1"><i><font color="Green" title="E28">A6</font></i></a></i>;<br/>
<a NAME="E12:35_1"/><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:35_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="lattice5.html#E5:35_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>
<a NAME="E13:35_1"/><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x2</font>,<font color="Maroon" title="c16">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E7:35_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="lattice5.html#E9:35_1"><i><font color="Green" title="E28">A6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">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="E15:35_1"/><i><font color="Green" title="E31">A9</font></i>: 
( <font color="Maroon" title="c17">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="c5">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="c17">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="c5">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="c18">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="E17:35_1"/><i><font color="Green" title="E32">A10</font></i>: 
( <font color="Maroon" title="c18">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="c6">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="c18">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="c6">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="E18:35_1"/>
( <font color="Maroon" title="c17">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">E2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">y1</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="c17">E1</font> )
 
<b>by </b><i><a class="txt" href="lattice5.html#E1:35_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="lattice5.html#E12:35_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="lattice5.html#E15:35_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="lattice5.html#E17:35_1"><i><font color="Green" title="E32">A10</font></i></a></i>;<br/>
<a NAME="E19:35_1"/><b>then </b><i><font color="Green" title="E33">A11</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E12:35_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="lattice5.html#E17:35_1"><i><font color="Green" title="E32">A10</font></i></a></i>;<br/>
<a NAME="E20:35_1"/>
( <font color="Maroon" title="c17">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">E2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c15">x2</font>,<font color="Maroon" title="c16">y2</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="c18">E2</font> )
 
<b>by </b><i><a class="txt" href="lattice5.html#E1:35_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="lattice5.html#E13:35_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="lattice5.html#E15:35_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="lattice5.html#E17:35_1"><i><font color="Green" title="E32">A10</font></i></a></i>;<br/>
<a NAME="E21:35_1"/><b>then </b>
<font color="Maroon" title="c6">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E13:35_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="lattice5.html#E15:35_1"><i><font color="Green" title="E31">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E22:35_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E19:35_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:35"/>
 <a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c3">d</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="waybel_1.html#D1" title="WAYBEL_1:def.1">WAYBEL_1:def 1</a></i>;<br/>


</div>
