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

<b>let </b><font color="Maroon" title="c1">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>set </b><font color="Maroon" title="c2">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>;<br/>
<b>set </b><font color="Maroon" title="c3">D</font> =  <a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E1:84"/><i><font color="Green" title="E63">A1</font></i>: 
 <a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<b>by </b><i><a class="ref" href="lattice5.html#T43" target="_self" title="LATTICE5:th.43">Th43</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">S</font> being    <a href="lattice5.html#M2" title="LATTICE5:mode.2">ExtensionSeq</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>, <a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font>;<br/>
<b>set </b><font color="Maroon" title="c5">FS</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> ;<br/>
<a NAME="E3:84"/><i><font color="Green" title="E64">A2</font></i>: 
<font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,<span class="p2">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="lattice5.html#D21" target="_self" title="LATTICE5:def.21">Def21</a></i>;<br/>
<a NAME="E4:84"/><b>then </b><i><font color="Green" title="E65">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <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="c1">L</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E5:84"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E6:84"/><b>then </b><i><font color="Green" title="E66">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E4:84"><i><font color="Green" title="E65">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">xx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:84"/><i><font color="Green" title="E67">A5</font></i>: 
<font color="Maroon" title="c6">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">FS</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  as  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>  <b>by </b><i><a class="txt" href="lattice5.html#E6:84"><i><font color="Green" title="E66">A4</font></i></a>, <a class="txt" href="lattice5.html#E8:84"><i><font color="Green" title="E67">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">FD</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  as   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c7">FS</font>,<font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="lattice5.html#T44" target="_self" title="LATTICE5:th.44">Th44</a></i>;<br/>
<a NAME="E11:84"/><i><font color="Green" title="E68">A6</font></i>: 
<font color="Maroon" title="c8">FD</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<a NAME="E1:84_1"/><i><font color="Green" title="E69">A7</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</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="c1">L</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E1:84"><i><font color="Green" title="E63">A1</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E2:84_1"/>
for <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> holds <br/> ex <font color="Olive" title="b2">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">z</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="c7">FS</font>,<font color="Maroon" title="c7">FS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">FD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:84_1_1"/>
<font color="Maroon" title="c9">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:84_1_1"/><i><font color="Green" title="E70">A8</font></i>: 
<font color="Maroon" title="c10">z</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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E4:84_1_1"/><i><font color="Green" title="E71">A9</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:84_1"><i><font color="Green" title="E69">A7</font></i></a>, <a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">z</font>
;<br/>

<a NAME="E5:84_1_1"/><i><font color="Green" title="E72">A10</font></i>: 
<font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,<span class="p2">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="lattice5.html#D21" target="_self" title="LATTICE5:def.21">Def21</a></i>;<br/>
<a NAME="E6:84_1_1"/><b>then </b><i><font color="Green" title="E73">A11</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 
<b>by </b><i><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E7:84_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E8:84_1_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">FS</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E6:84_1_1"><i><font color="Green" title="E73">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E9:84_1_1"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">FS</font>,<font color="Maroon" title="c7">FS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E10:84_1_1"/>
<font color="Maroon" title="c10">z</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="c7">FS</font>,<font color="Maroon" title="c7">FS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:84_1_1"><i><font color="Green" title="E70">A8</font></i></a></i>;<br/>

<a NAME="E11:84_1_1"/><i><font color="Green" title="E74">A12</font></i>: 
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:84_1_1"><i><font color="Green" title="E70">A8</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:84_1_1"/><i><font color="Green" title="E75">A13</font></i>: 
 <a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E5:84_1_1"><i><font color="Green" title="E72">A10</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<a NAME="E13:84_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E14:84_1_1"/><b>then </b>
 <a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c8">FD</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E12:84_1_1"><i><font color="Green" title="E75">A13</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E15:84_1_1"/>
<font color="Maroon" title="c9">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">FD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:84_1_1"><i><font color="Green" title="E71">A9</font></i></a>, <a class="txt" href="lattice5.html#E11:84_1_1"><i><font color="Green" title="E74">A12</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:84_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">FD</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="c1">L</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T16" title="FUNCT_2:th.16">FUNCT_2:16</a></i>;<br/>
<b>hence </b><a NAME="E4:84_1"/>
<font color="Maroon" title="c8">FD</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 <b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:84"/>
 <a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c8">FD</font> is <a href="waybel_0.html#V20" title="WAYBEL_0:attr.20">join-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</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#D35" title="WAYBEL_0:def.35">WAYBEL_0:def 35</a><br/>

<b>set </b><font color="Maroon" title="c11">f</font> =  <a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c8">FD</font>;<br/>
<a NAME="E1:84_2"/><i><font color="Green" title="E69">A14</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c8">FD</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="c9">a</font>,<font color="Maroon" title="c10">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="c7">FS</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="E2:84_2"/><i><font color="Green" title="E70">A15</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="c8">FD</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="c1">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="c12">e1</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c7">FS</font><b> such that </b><br/><a NAME="E4:84_2"/><i><font color="Green" title="E71">A16</font></i>: 
<font color="Maroon" title="c12">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="c8">FD</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font>
 <b>and </b><br/><a NAME="E5:84_2"/><i><font color="Green" title="E72">A17</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="c7">FS</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="c12">e1</font> iff <font color="Maroon" title="c8">FD</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="c9">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="c13">e2</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c7">FS</font><b> such that </b><br/><a NAME="E7:84_2"/><i><font color="Green" title="E73">A18</font></i>: 
<font color="Maroon" title="c13">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="c8">FD</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font>
 <b>and </b><br/><a NAME="E8:84_2"/><i><font color="Green" title="E74">A19</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="c7">FS</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="c13">e2</font> iff <font color="Maroon" title="c8">FD</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="c10">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="c14">e3</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c7">FS</font><b> such that </b><br/><a NAME="E10:84_2"/><i><font color="Green" title="E75">A20</font></i>: 
<font color="Maroon" title="c14">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="c8">FD</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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>and </b><br/><a NAME="E11:84_2"/><i><font color="Green" title="E76">A21</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="c7">FS</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="c14">e3</font> iff <font color="Maroon" title="c8">FD</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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">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="E12:84_2"/><i><font color="Green" title="E77">A22</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c12">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">FS</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97" title="ORDERS_1:th.97">ORDERS_1:97</a></i>;<br/>
<a NAME="E13:84_2"/><i><font color="Green" title="E78">A23</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c13">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">FS</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97" title="ORDERS_1:th.97">ORDERS_1:97</a></i>;<br/>
<a NAME="E14:84_2"/><i><font color="Green" title="E79">A24</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c14">e3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">FS</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97" title="ORDERS_1:th.97">ORDERS_1:97</a></i>;<br/>
<a NAME="E15:84_2"/><i><font color="Green" title="E80">A25</font></i>: 
<font color="Maroon" title="c12">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c13">e2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c14">e3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c16">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:84_2_1_1"/><i><font color="Green" title="E81">A26</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">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="c12">e1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c17">x'</font> = <font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c18">y'</font> = <font color="Maroon" title="c16">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">FS</font> <b>by </b><i><a class="txt" href="lattice5.html#E12:84_2"><i><font color="Green" title="E77">A22</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/><a NAME="E3:84_2_1_1"/><i><font color="Green" title="E82">A27</font></i>: 
<font color="Maroon" title="c8">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">x'</font>,<font color="Maroon" title="c18">y'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E5:84_2"><i><font color="Green" title="E72">A17</font></i></a>, <a class="txt" href="lattice5.html#E1:84_2_1_1"><i><font color="Green" title="E81">A26</font></i></a></i>;<br/><a NAME="E4:84_2_1_1"/>
<font color="Maroon" title="c9">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><a NAME="E5:84_2_1_1"/><b>then </b>
<font color="Maroon" title="c8">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">x'</font>,<font color="Maroon" title="c18">y'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:84_2_1_1"><i><font color="Green" title="E82">A27</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E6:84_2_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">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="c14">e3</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E11:84_2"><i><font color="Green" title="E76">A21</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:84_2_1"/><b>then </b><i><font color="Green" title="E81">A28</font></i>: 
<font color="Maroon" title="c12">e1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c14">e3</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c16">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:84_2_1_2"/><i><font color="Green" title="E82">A29</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">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="c13">e2</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c17">x'</font> = <font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c18">y'</font> = <font color="Maroon" title="c16">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">FS</font> <b>by </b><i><a class="txt" href="lattice5.html#E13:84_2"><i><font color="Green" title="E78">A23</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/><a NAME="E3:84_2_1_2"/><i><font color="Green" title="E83">A30</font></i>: 
<font color="Maroon" title="c8">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">x'</font>,<font color="Maroon" title="c18">y'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E8:84_2"><i><font color="Green" title="E74">A19</font></i></a>, <a class="txt" href="lattice5.html#E1:84_2_1_2"><i><font color="Green" title="E82">A29</font></i></a></i>;<br/><a NAME="E4:84_2_1_2"/>
<font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><a NAME="E5:84_2_1_2"/><b>then </b>
<font color="Maroon" title="c8">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">x'</font>,<font color="Maroon" title="c18">y'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:84_2_1_2"><i><font color="Green" title="E83">A30</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E6:84_2_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">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="c14">e3</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E11:84_2"><i><font color="Green" title="E76">A21</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:84_2_1"/><b>then </b>
<font color="Maroon" title="c13">e2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c14">e3</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a></i>;<br/>
<a NAME="E5:84_2_1"/><b>then </b>
<font color="Maroon" title="c12">e1</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c13">e2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c14">e3</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E2:84_2_1"><i><font color="Green" title="E81">A28</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E6:84_2_1"/>
<font color="Maroon" title="c12">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c13">e2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c14">e3</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:84_2"/><i><font color="Green" title="E81">A31</font></i>: 
<font color="Maroon" title="c14">e3</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c12">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c13">e2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:84_2_2"/>
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</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="c14">e3</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</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="c12">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c13">e2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">u</font>, <font color="Maroon" title="c16">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:84_2_2_1"/><i><font color="Green" title="E82">A32</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">v</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="c14">e3</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c17">x</font> = <font color="Maroon" title="c15">u</font>, <font color="Maroon" title="c18">y</font> = <font color="Maroon" title="c16">v</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">FS</font> <b>by </b><i><a class="txt" href="lattice5.html#E14:84_2"><i><font color="Green" title="E79">A24</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<a NAME="E3:84_2_2_1"/><i><font color="Green" title="E83">A33</font></i>: 
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">FS</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E14:84_2"><i><font color="Green" title="E79">A24</font></i></a>, <a class="txt" href="lattice5.html#E1:84_2_2_1"><i><font color="Green" title="E82">A32</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<a NAME="E4:84_2_2_1"/>
<font color="Maroon" title="c8">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E11:84_2"><i><font color="Green" title="E76">A21</font></i></a>, <a class="txt" href="lattice5.html#E1:84_2_2_1"><i><font color="Green" title="E82">A32</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">z1</font>, <font color="Maroon" title="c20">z2</font>, <font color="Maroon" title="c21">z3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">FS</font><b> such that </b><br/><a NAME="E6:84_2_2_1"/><i><font color="Green" title="E84">A34</font></i>: 
<font color="Maroon" title="c8">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c19">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <b>and </b><br/><a NAME="E7:84_2_2_1"/><i><font color="Green" title="E85">A35</font></i>: 
<font color="Maroon" title="c8">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c20">z2</font>,<font color="Maroon" title="c21">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <b>and </b><br/><a NAME="E8:84_2_2_1"/><i><font color="Green" title="E86">A36</font></i>: 
<font color="Maroon" title="c8">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c19">z1</font>,<font color="Maroon" title="c20">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <b>and </b><br/><a NAME="E9:84_2_2_1"/><i><font color="Green" title="E87">A37</font></i>: 
<font color="Maroon" title="c8">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c21">z3</font>,<font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="lattice5.html#T45" target="_self" title="LATTICE5:th.45">Th45</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x</font> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">z1</font> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">z2</font> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">z3</font> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font> ) );<br/>
<a NAME="E10:84_2_2_1"/><i><font color="Green" title="E88">A38</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">w1</font>, <font color="Olive" title="b3">w2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">w1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Olive" title="b3">w2</font>] holds <br/><font color="Olive" title="b2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">w2</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E68"><i><font color="Green" title="E58">Lm3</font></i></a></i>;<br/>
<a NAME="E11:84_2_2_1"/><i><font color="Green" title="E89">A39</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5 holds <br/> ex <font color="Olive" title="b2">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">w</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:84_2_2_1_1"/><i><font color="Green" title="E90">A40</font></i>: 
<font color="Maroon" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_2_1_1_1"/>
( <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 )
 </span><b>by </b><i><a class="txt" href="lattice5.html#E1:84_2_2_1_1"><i><font color="Green" title="E90">A40</font></i></a>, <a class="txt" href="lattice5.html#E68"><i><font color="Green" title="E58">Lm3</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:84_2_2_1_1_1_1"/><i><font color="Green" title="E91">A41</font></i>: 
<font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c23">w</font> = <font color="Maroon" title="c17">x</font>;<br/><b>thus </b><a NAME="E2:84_2_2_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c22">m</font>,<font color="Maroon" title="c23">w</font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:84_2_2_1_1_1_1"><i><font color="Green" title="E91">A41</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:84_2_2_1_1_1_2"/><i><font color="Green" title="E91">A42</font></i>: 
<font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c23">w</font> = <font color="Maroon" title="c19">z1</font>;<br/><b>thus </b><a NAME="E2:84_2_2_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c22">m</font>,<font color="Maroon" title="c23">w</font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:84_2_2_1_1_1_2"><i><font color="Green" title="E91">A42</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_1_1_3"><b>suppose </b></a><a NAME="E1:84_2_2_1_1_1_3"/><i><font color="Green" title="E91">A43</font></i>: 
<font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c23">w</font> = <font color="Maroon" title="c20">z2</font>;<br/><b>thus </b><a NAME="E2:84_2_2_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c22">m</font>,<font color="Maroon" title="c23">w</font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:84_2_2_1_1_1_3"><i><font color="Green" title="E91">A43</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_1_1_4"><b>suppose </b></a><a NAME="E1:84_2_2_1_1_1_4"/><i><font color="Green" title="E91">A44</font></i>: 
<font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c23">w</font> = <font color="Maroon" title="c21">z3</font>;<br/><b>thus </b><a NAME="E2:84_2_2_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c22">m</font>,<font color="Maroon" title="c23">w</font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:84_2_2_1_1_1_4"><i><font color="Green" title="E91">A44</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_1_1_5"><b>suppose </b></a><a NAME="E1:84_2_2_1_1_1_5"/><i><font color="Green" title="E91">A45</font></i>: 
<font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c23">w</font> = <font color="Maroon" title="c18">y</font>;<br/><b>thus </b><a NAME="E2:84_2_2_1_1_1_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c22">m</font>,<font color="Maroon" title="c23">w</font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:84_2_2_1_1_1_5"><i><font color="Green" title="E91">A45</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E12:84_2_2_1"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5 &amp; ( for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">k</font>,<font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font>] ) )
 
<b>from </b><i><a class="ref" href="finseq_1.html#S1" title="FINSEQ_1:sch.1">FINSEQ_1:sch 1</a>(<a class="txt" href="lattice5.html#E10:84_2_2_1"><i><font color="Green" title="E88">A38</font></i></a>, <a class="txt" href="lattice5.html#E11:84_2_2_1"><i><font color="Green" title="E89">A39</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c22">h</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E14:84_2_2_1"/><i><font color="Green" title="E90">A46</font></i>: 
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c22">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5
 <b>and </b><br/><a NAME="E15:84_2_2_1"/><i><font color="Green" title="E91">A47</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5 holds <br/>( ( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x</font> ) &amp; ( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">z1</font> ) &amp; ( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 implies <font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">z2</font> ) &amp; ( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 implies <font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">z3</font> ) &amp; ( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 implies <font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font> ) )
 ;<br/>
<a NAME="E16:84_2_2_1"/><i><font color="Green" title="E92">A48</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5 &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5 &amp; 3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5 &amp; 4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5 &amp; 5 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5 )
 
;<br/>
<a NAME="E17:84_2_2_1"/><i><font color="Green" title="E93">A49</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 
<b>by </b><i><a class="txt" href="lattice5.html#E14:84_2_2_1"><i><font color="Green" title="E90">A46</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E18:84_2_2_1"/><i><font color="Green" title="E94">A50</font></i>: 
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="lattice5.html#E15:84_2_2_1"><i><font color="Green" title="E91">A47</font></i></a>, <a class="txt" href="lattice5.html#E16:84_2_2_1"><i><font color="Green" title="E92">A48</font></i></a></i>;<br/>
<i><font color="Green" title="E95">A51</font></i>: <font color="Maroon" title="c16">v</font> = 
<font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5
<b>by </b><i><a class="txt" href="lattice5.html#E15:84_2_2_1"><i><font color="Green" title="E91">A47</font></i></a>, <a class="txt" href="lattice5.html#E16:84_2_2_1"><i><font color="Green" title="E92">A48</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">h</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E14:84_2_2_1"><i><font color="Green" title="E90">A46</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E20:84_2_2_1"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">h</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></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="c12">e1</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c13">e2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:84_2_2_1_3"/><i><font color="Green" title="E96">A52</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">j</font> &amp; <font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">h</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_2_1_3_1"/>
( <font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 </span><b>by </b><i><a class="txt" href="lattice5.html#E17:84_2_2_1"><i><font color="Green" title="E93">A49</font></i></a>, <a class="txt" href="lattice5.html#E1:84_2_2_1_3"><i><font color="Green" title="E96">A52</font></i></a>, <a class="txt" href="lattice5.html#E67"><i><font color="Green" title="E57">Lm2</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_3_1_1"><b>suppose </b></a><a NAME="E1:84_2_2_1_3_1_1"/><i><font color="Green" title="E97">A53</font></i>: 
<font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:84_2_2_1_3_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c19">z1</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="c12">e1</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E5:84_2"><i><font color="Green" title="E72">A17</font></i></a>, <a class="txt" href="lattice5.html#E6:84_2_2_1"><i><font color="Green" title="E84">A34</font></i></a></i>;<br/><a NAME="E3:84_2_2_1_3_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c19">z1</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="c12">e1</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E15:84_2_2_1"><i><font color="Green" title="E91">A47</font></i></a>, <a class="txt" href="lattice5.html#E16:84_2_2_1"><i><font color="Green" title="E92">A48</font></i></a></i>;<br/><a NAME="E4:84_2_2_1_3_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span></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="c12">e1</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E15:84_2_2_1"><i><font color="Green" title="E91">A47</font></i></a>, <a class="txt" href="lattice5.html#E16:84_2_2_1"><i><font color="Green" title="E92">A48</font></i></a></i>;<br/><b>hence </b><a NAME="E5:84_2_2_1_3_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></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="c12">e1</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c13">e2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:84_2_2_1_3_1_1"><i><font color="Green" title="E97">A53</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_3_1_2"><b>suppose </b></a><a NAME="E1:84_2_2_1_3_1_2"/><i><font color="Green" title="E97">A54</font></i>: 
<font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:84_2_2_1_3_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">z2</font>,<font color="Maroon" title="c21">z3</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="c12">e1</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E5:84_2"><i><font color="Green" title="E72">A17</font></i></a>, <a class="txt" href="lattice5.html#E7:84_2_2_1"><i><font color="Green" title="E85">A35</font></i></a></i>;<br/><a NAME="E3:84_2_2_1_3_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c21">z3</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="c12">e1</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E15:84_2_2_1"><i><font color="Green" title="E91">A47</font></i></a>, <a class="txt" href="lattice5.html#E16:84_2_2_1"><i><font color="Green" title="E92">A48</font></i></a></i>;<br/><a NAME="E4:84_2_2_1_3_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 4</span>)</span></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="c12">e1</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E15:84_2_2_1"><i><font color="Green" title="E91">A47</font></i></a>, <a class="txt" href="lattice5.html#E16:84_2_2_1"><i><font color="Green" title="E92">A48</font></i></a></i>;<br/><b>hence </b><a NAME="E5:84_2_2_1_3_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></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="c12">e1</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c13">e2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:84_2_2_1_3_1_2"><i><font color="Green" title="E97">A54</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_3_1_3"><b>suppose </b></a><a NAME="E1:84_2_2_1_3_1_3"/><i><font color="Green" title="E97">A55</font></i>: 
<font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:84_2_2_1_3_1_3"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">z1</font>,<font color="Maroon" title="c20">z2</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="c13">e2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E8:84_2"><i><font color="Green" title="E74">A19</font></i></a>, <a class="txt" href="lattice5.html#E8:84_2_2_1"><i><font color="Green" title="E86">A36</font></i></a></i>;<br/><a NAME="E3:84_2_2_1_3_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c20">z2</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="c13">e2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E15:84_2_2_1"><i><font color="Green" title="E91">A47</font></i></a>, <a class="txt" href="lattice5.html#E16:84_2_2_1"><i><font color="Green" title="E92">A48</font></i></a></i>;<br/><a NAME="E4:84_2_2_1_3_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span></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="c13">e2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E15:84_2_2_1"><i><font color="Green" title="E91">A47</font></i></a>, <a class="txt" href="lattice5.html#E16:84_2_2_1"><i><font color="Green" title="E92">A48</font></i></a></i>;<br/><b>hence </b><a NAME="E5:84_2_2_1_3_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></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="c12">e1</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c13">e2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:84_2_2_1_3_1_3"><i><font color="Green" title="E97">A55</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_3_1_4"><b>suppose </b></a><a NAME="E1:84_2_2_1_3_1_4"/><i><font color="Green" title="E97">A56</font></i>: 
<font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ;<br/><div class="add"><a NAME="E2:84_2_2_1_3_1_4"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">z3</font>,<font color="Maroon" title="c18">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="c13">e2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E8:84_2"><i><font color="Green" title="E74">A19</font></i></a>, <a class="txt" href="lattice5.html#E9:84_2_2_1"><i><font color="Green" title="E87">A37</font></i></a></i>;<br/><a NAME="E3:84_2_2_1_3_1_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c18">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="c13">e2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E15:84_2_2_1"><i><font color="Green" title="E91">A47</font></i></a>, <a class="txt" href="lattice5.html#E16:84_2_2_1"><i><font color="Green" title="E92">A48</font></i></a></i>;<br/><a NAME="E4:84_2_2_1_3_1_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 4</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5</span>)</span></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="c13">e2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E15:84_2_2_1"><i><font color="Green" title="E91">A47</font></i></a>, <a class="txt" href="lattice5.html#E16:84_2_2_1"><i><font color="Green" title="E92">A48</font></i></a></i>;<br/><b>hence </b><a NAME="E5:84_2_2_1_3_1_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></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="c12">e1</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c13">e2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:84_2_2_1_3_1_4"><i><font color="Green" title="E97">A56</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E21:84_2_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">v</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="c12">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c13">e2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:84_2_2_1"><i><font color="Green" title="E83">A33</font></i></a>, <a class="txt" href="lattice5.html#E17:84_2_2_1"><i><font color="Green" title="E93">A49</font></i></a>, <a class="txt" href="lattice5.html#E18:84_2_2_1"><i><font color="Green" title="E94">A50</font></i></a>, <a class="txt" href="lattice5.html#E19:84_2_2_1"><i><font color="Green" title="E95">A51</font></i></a>, <a class="ref" href="eqrel_1.html#T36" title="EQREL_1:th.36">EQREL_1:36</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:84_2_2"/>
<font color="Maroon" title="c14">e3</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c12">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c13">e2</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</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="c8">FD</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="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> = 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</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="c8">FD</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">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="c8">FD</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">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#E2:84_2"><i><font color="Green" title="E70">A15</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="c8">FD</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</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="c8">FD</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#T41" title="YELLOW_0:th.41">YELLOW_0:41</a></i>
<br/>.= 
<font color="Maroon" title="c12">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c13">e2</font>
<b>by </b><i><a class="txt" href="lattice5.html#E4:84_2"><i><font color="Green" title="E71">A16</font></i></a>, <a class="txt" href="lattice5.html#E7:84_2"><i><font color="Green" title="E73">A18</font></i></a>, <a class="ref" href="lattice5.html#T10" target="_self" title="LATTICE5:th.10">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c8">FD</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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E10:84_2"><i><font color="Green" title="E75">A20</font></i></a>, <a class="txt" href="lattice5.html#E15:84_2"><i><font color="Green" title="E80">A25</font></i></a>, <a class="txt" href="lattice5.html#E16:84_2"><i><font color="Green" title="E81">A31</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c8">FD</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#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">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#T41" title="YELLOW_0:th.41">YELLOW_0:41</a></i>
;<br/>
<b>hence </b><a NAME="E18:84_2"/>
 <a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c8">FD</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">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#E1:84_2"><i><font color="Green" title="E69">A14</font></i></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">f</font> =  <a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c8">FD</font> as   <a href="lattice5.html#NM1" title="LATTICE5:NM.1">Homomorphism</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c7">FS</font></span>)</span> <b>by </b><i><a class="ref" href="lattice5.html#T16" target="_self" title="LATTICE5:th.16">Th16</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">FS</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c9">f</font>
;<br/>

<b>thus </b><a NAME="E14:84"/>
<font color="Maroon" title="c9">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="lattice5.html#E11:84"><i><font color="Green" title="E68">A6</font></i></a>, <a class="ref" href="lattice5.html#T17" target="_self" title="LATTICE5:th.17">Th17</a></i>;<br/>

<a NAME="E15:84"/><i><font color="Green" title="E69">A57</font></i>: 
 <a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_2.html#D2" title="YELLOW_2:def.2">YELLOW_2:def 2</a></i>;<br/>
<a NAME="E16:84"/><i><font color="Green" title="E70">A58</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f</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="c1">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/>
<a NAME="E17:84"/><i><font color="Green" title="E71">A59</font></i>: 
 ex <font color="Olive" title="b1">e</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c7">FS</font> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c9">f</font></span>)</span> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c7">FS</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c10">A'</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> , <font color="Maroon" title="c11">d'</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c10">A'</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c12">Aq'</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> , <font color="Maroon" title="c13">dq'</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c12">Aq'</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E2:84_3"/><i><font color="Green" title="E72">A60</font></i>: 
<font color="Maroon" title="c12">Aq'</font>,<font color="Maroon" title="c13">dq'</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c10">A'</font>,<font color="Maroon" title="c11">d'</font>
 <b>and </b><br/><a NAME="E3:84_3"/><i><font color="Green" title="E73">A61</font></i>: 
( <font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <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="c10">A'</font>,<font color="Maroon" title="c11">d'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c12">Aq'</font>,<font color="Maroon" title="c13">dq'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice5.html#D21" target="_self" title="LATTICE5:def.21">Def21</a></i>;<br/>
<a NAME="E4:84_3"/>
( <font color="Maroon" title="c10">A'</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="c1">L</font> &amp; <font color="Maroon" title="c11">d'</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:84"><i><font color="Green" title="E64">A2</font></i></a>, <a class="txt" href="lattice5.html#E3:84_3"><i><font color="Green" title="E73">A61</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">q</font> being    <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of  <a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E6:84_3"/><i><font color="Green" title="E74">A62</font></i>: 
<font color="Maroon" title="c12">Aq'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice5.html#K14" title="LATTICE5:func.14">NextSet</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <b>and </b><br/><a NAME="E7:84_3"/><i><font color="Green" title="E75">A63</font></i>: 
<font color="Maroon" title="c13">dq'</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lattice5.html#K16" title="LATTICE5:func.16">NextDelta</a> <font color="Maroon" title="c14">q</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E2:84_3"><i><font color="Green" title="E72">A60</font></i></a>, <a class="ref" href="lattice5.html#D20" target="_self" title="LATTICE5:def.20">Def20</a></i>;<br/>
<a NAME="E8:84_3"/><i><font color="Green" title="E76">A64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice5.html#K16" title="LATTICE5:func.16">NextDelta</a> <font color="Maroon" title="c14">q</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:84_3"><i><font color="Green" title="E73">A61</font></i></a>, <a class="txt" href="lattice5.html#E7:84_3"><i><font color="Green" title="E75">A63</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:84_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E10:84_3"/><b>then </b><i><font color="Green" title="E77">A65</font></i>: 
 <a href="lattice5.html#K16" title="LATTICE5:func.16">NextDelta</a> <font color="Maroon" title="c14">q</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c8">FD</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E8:84_3"><i><font color="Green" title="E76">A64</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E11:84_3"/><i><font color="Green" title="E78">A66</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice5.html#K14" title="LATTICE5:func.14">NextSet</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:84_3"><i><font color="Green" title="E73">A61</font></i></a>, <a class="txt" href="lattice5.html#E6:84_3"><i><font color="Green" title="E74">A62</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E12:84_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E13:84_3"/><b>then </b><i><font color="Green" title="E79">A67</font></i>: 
 <a href="lattice5.html#K14" title="LATTICE5:func.14">NextSet</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">FS</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E11:84_3"><i><font color="Green" title="E78">A66</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E14:84_3"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E69"><i><font color="Green" title="E59">Lm4</font></i></a></i>;<br/>
<a NAME="E15:84_3"/><b>then </b>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></i>;<br/>
<a NAME="E16:84_3"/><b>then </b><i><font color="Green" title="E80">A68</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">q</font>
 
<b>by </b><i><a class="ref" href="lattice5.html#T28" target="_self" title="LATTICE5:th.28">Th28</a></i>;<br/>
<a NAME="E17:84_3"/><b>then </b>
<font color="Maroon" title="c14">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">q</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E18:84_3"/><b>then </b>
<font color="Maroon" title="c14">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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">u</font>,<font color="Olive" title="b2">v</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">u</font>, <font color="Olive" title="b2">v</font> is    <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>, <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="c1">L</font> : <span class="p1">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</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> 
 
<b>by </b><i><a class="ref" href="lattice5.html#D14" target="_self" title="LATTICE5:def.14">Def14</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">u</font>, <font color="Maroon" title="c16">v</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>, <font color="Maroon" title="c17">a</font>, <font color="Maroon" title="c18">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E20:84_3"/><i><font color="Green" title="E81">A69</font></i>: 
( <font color="Maroon" title="c14">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">v</font>,<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c18">b</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span> &amp; <span class="p1">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c18">b</font> )
 ;<br/>
<a NAME="E21:84_3"/>
 <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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="c1">L</font>
 
<b>by </b><i><a class="ref" href="lattice5.html#T24" target="_self" title="LATTICE5:th.24">Th24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">Q</font> =  <a href="lattice5.html#K10" title="LATTICE5:func.10">Quadr</a> <font color="Maroon" title="c14">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span> ;<br/>
<a NAME="E23:84_3"/><i><font color="Green" title="E82">A70</font></i>: 
 <a href="lattice5.html#K10" title="LATTICE5:func.10">Quadr</a> <font color="Maroon" title="c14">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">v</font>,<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c18">b</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E16:84_3"><i><font color="Green" title="E80">A68</font></i></a>, <a class="txt" href="lattice5.html#E20:84_3"><i><font color="Green" title="E81">A69</font></i></a>, <a class="ref" href="lattice5.html#D15" target="_self" title="LATTICE5:def.15">Def15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">e</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c7">FS</font><b> such that </b><br/><a NAME="E25:84_3"/><i><font color="Green" title="E83">A71</font></i>: 
<font color="Maroon" title="c20">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font>
 <b>and </b><br/><a NAME="E26:84_3"/><i><font color="Green" title="E84">A72</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="c7">FS</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="c20">e</font> iff <font color="Maroon" title="c8">FD</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="c18">b</font> )
 <b>by </b><i><a class="ref" href="lattice5.html#D9" target="_self" title="LATTICE5:def.9">Def9</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c20">e</font>
;<br/>

<a NAME="E27:84_3"/>
<font color="Maroon" title="c20">e</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> <font color="Maroon" title="c9">f</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E16:84"><i><font color="Green" title="E70">A58</font></i></a>, <a class="txt" href="lattice5.html#E25:84_3"><i><font color="Green" title="E83">A71</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E28:84_3"/>
<font color="Maroon" title="c20">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c9">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E15:84"><i><font color="Green" title="E69">A57</font></i></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>

<i><font color="Green" title="E85">A73</font></i>:  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> = 
 <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <span class="p1">(<span class="default"><a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#T24" target="_self" title="LATTICE5:th.24">Th24</a></i>
<br/>.= 
 <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#T25" target="_self" title="LATTICE5:th.25">Th25</a></i>
;<br/>
<a NAME="E30:84_3"/><i><font color="Green" title="E86">A74</font></i>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E69"><i><font color="Green" title="E59">Lm4</font></i></a></i>;<br/>
<a NAME="E31:84_3"/><b>then </b><i><font color="Green" title="E87">A75</font></i>: 
 <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K14" title="LATTICE5:func.14">NextSet</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E29:84_3"><i><font color="Green" title="E85">A73</font></i></a>, <a class="ref" href="lattice5.html#T32" target="_self" title="LATTICE5:th.32">Th32</a></i>;<br/>
<a NAME="E32:84_3"/><i><font color="Green" title="E88">A76</font></i>: 
(  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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="c1">L</font> &amp;  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c14">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="ref" href="lattice5.html#T24" target="_self" title="LATTICE5:th.24">Th24</a>, <a class="ref" href="lattice5.html#T29" target="_self" title="LATTICE5:th.29">Th29</a></i>;<br/>
 <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c14">q</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> = 
 <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K11" title="LATTICE5:func.11">BiFun</a> <span class="p2">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c14">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice5.html#K10" title="LATTICE5:func.10">Quadr</a> <font color="Maroon" title="c14">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#T30" target="_self" title="LATTICE5:th.30">Th30</a></i>
<br/>.= 
 <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c19">Q</font>
<b>by </b><i><a class="txt" href="lattice5.html#E32:84_3"><i><font color="Green" title="E88">A76</font></i></a>, <a class="ref" href="lattice5.html#D16" target="_self" title="LATTICE5:def.16">Def16</a></i>
;<br/>
<a NAME="E34:84_3"/><b>then </b>
 <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c19">Q</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice5.html#K16" title="LATTICE5:func.16">NextDelta</a> <font color="Maroon" title="c14">q</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E30:84_3"><i><font color="Green" title="E86">A74</font></i></a>, <a class="ref" href="lattice5.html#T35" target="_self" title="LATTICE5:th.35">Th35</a></i>;<br/>
<a NAME="E35:84_3"/><b>then </b><i><font color="Green" title="E89">A77</font></i>: 
 <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c19">Q</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c8">FD</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E10:84_3"><i><font color="Green" title="E77">A65</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E36:84_3"/><i><font color="Green" title="E90">A78</font></i>: 
 <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">FS</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E13:84_3"><i><font color="Green" title="E79">A67</font></i></a>, <a class="txt" href="lattice5.html#E31:84_3"><i><font color="Green" title="E87">A75</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E37:84_3"/><i><font color="Green" title="E91">A79</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#K7" title="LATTICE5:func.7">new_bi_fun</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c19">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E38:84_3"/><i><font color="Green" title="E92">A80</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E39:84_3"/><b>then </b><i><font color="Green" title="E93">A81</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E40:84_3"/><i><font color="Green" title="E94">A82</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E38:84_3"><i><font color="Green" title="E92">A80</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E41:84_3"/><i><font color="Green" title="E95">A83</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E42:84_3"/><b>then </b><i><font color="Green" title="E96">A84</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E43:84_3"/><i><font color="Green" title="E97">A85</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E41:84_3"><i><font color="Green" title="E95">A83</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E44:84_3"/><i><font color="Green" title="E98">A86</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c19">Q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E37:84_3"><i><font color="Green" title="E91">A79</font></i></a>, <a class="txt" href="lattice5.html#E40:84_3"><i><font color="Green" title="E94">A82</font></i></a>, <a class="txt" href="lattice5.html#E42:84_3"><i><font color="Green" title="E96">A84</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">W</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>, <font color="Maroon" title="c22">V</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">FS</font> <b>by </b><i><a class="txt" href="lattice5.html#E36:84_3"><i><font color="Green" title="E90">A78</font></i></a>, <a class="txt" href="lattice5.html#E39:84_3"><i><font color="Green" title="E93">A81</font></i></a>, <a class="txt" href="lattice5.html#E43:84_3"><i><font color="Green" title="E97">A85</font></i></a></i>;<br/>
<font color="Maroon" title="c8">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c21">W</font>,<font color="Maroon" title="c22">V</font> = 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c19">Q</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E35:84_3"><i><font color="Green" title="E89">A77</font></i></a>, <a class="txt" href="lattice5.html#E44:84_3"><i><font color="Green" title="E98">A86</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c19">Q</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> 
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c18">b</font>
<b>by </b><i><a class="txt" href="lattice5.html#E23:84_3"><i><font color="Green" title="E82">A70</font></i></a>, <a class="ref" href="mcart_1.html#D11" title="MCART_1:def.11">MCART_1:def 11</a></i>
;<br/>
<a NAME="E47:84_3"/><b>then </b><i><font color="Green" title="E99">A87</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c20">e</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E26:84_3"><i><font color="Green" title="E84">A72</font></i></a></i>;<br/>
<a NAME="E48:84_3"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_3_4"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:84_3_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:84_3_4"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E49:84_3"/>
<font color="Maroon" title="c20">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c7">FS</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E47:84_3"><i><font color="Green" title="E99">A87</font></i></a>, <a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:84"/>
for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c7">FS</font><br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c9">f</font></span>)</span> &amp; <font color="Olive" title="b2">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c9">f</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">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="Olive" title="b1">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Olive" title="b2">e2</font> holds <br/> ex <font color="Olive" title="b5">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c7">FS</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 &amp; <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b5">F</font>,<font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font> )
 
<b>by </b><i><a class="ref" href="lattice5.html#T46" target="_self" title="LATTICE5:th.46">Th46</a></i>;<br/>
<b>hence </b><a NAME="E19:84"/>
 <a href="lattice5.html#K3" title="LATTICE5:func.3">type_of</a> <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3
 <b>by </b><i><a class="txt" href="lattice5.html#E17:84"><i><font color="Green" title="E71">A59</font></i></a>, <a class="ref" href="lattice5.html#T15" target="_self" title="LATTICE5:th.15">Th15</a></i>;<br/>


</div>
