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

<span class="kw">thus </span><a NAME="E1:9_1_1"/>
( <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> <a href="filter_1.html#R1" title="FILTER_1:pred.1">are_isomorphic</a>  implies  ex <font color="Olive" title="b1">f</font> being    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> st <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">f</font> being    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> st <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  implies <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> <a href="filter_1.html#R1" title="FILTER_1:pred.1">are_isomorphic</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c3">R</font> =  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">S</font> =  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font>;<br/>
<span class="kw">assume </span><a NAME="E1:9_1_1_1"/>
<font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> <a href="filter_1.html#R1" title="FILTER_1:pred.1">are_isomorphic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> st <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/>

<a NAME="E2:9_1_1_1"/><span class="kw">then </span>
 <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font>, <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#D9" title="FILTER_1:def.9">FILTER_1:def 9</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E4:9_1_1_1"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c5">F</font> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font>, <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#D8" title="WELLORD1:def.8">WELLORD1:def 8</a></span>;<br/>
<a NAME="E5:9_1_1_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<a NAME="E6:9_1_1_1"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
(  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<a NAME="E7:9_1_1_1"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</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">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">F</font> = <font color="Maroon" title="c5">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E6:9_1_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="funct_2.html#T1" title="FUNCT_2:th.1">FUNCT_2:1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">b1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font></span>)</span> &amp; <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font></span>)</span> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">a19</font> = <font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c10">b19</font> = <font color="Maroon" title="c8">b1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> ;<br/><a NAME="E2:9_1_1_1_1"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
<font color="Maroon" title="c6">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_1_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/><span class="kw">thus </span><a NAME="E3:9_1_1_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:9_1_1_1_1_1"/>
<font color="Maroon" title="c10">b19</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">a19</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c10">b19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T5" title="LATTICES:th.5">LATTICES:5</a></span>;<br/>
<a NAME="E2:9_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c8">b1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">b1</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E3:9_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">b1</font></span>)</span></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<a NAME="E4:9_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A6</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b19</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a19</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c10">b19</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">k1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font><span class="kw"> such that </span><br/><a NAME="E6:9_1_1_1_1_1"/><span class="lab"><font color="Green" title="E11">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1_1_1_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="lattice4.html#T6" target="_self" title="LATTICE4:th.6">Th9</a></span>;<br/>
<a NAME="E7:9_1_1_1_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="ref" href="lattices.html#T5" title="LATTICES:th.5">LATTICES:5</a></span>;<br/>
<a NAME="E8:9_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k1</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E9:9_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c11">k1</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E7:9_1_1_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<a NAME="E10:9_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A8</font></span>: 
<font color="Maroon" title="c8">b1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E11:9_1_1_1_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="ref" href="lattices.html#T5" title="LATTICES:th.5">LATTICES:5</a></span>;<br/>
<a NAME="E12:9_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k1</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E13:9_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c11">k1</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E7:9_1_1_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<a NAME="E14:9_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">a1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E15:9_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">a1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">b1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:9_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="filter_0.html#T6" title="FILTER_0:th.6">FILTER_0:6</a></span>;<br/>
<a NAME="E16:9_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">b1</font></span>)</span>,<font color="Maroon" title="c11">k1</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E17:9_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">b1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k1</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<a NAME="E18:9_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A9</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E19:9_1_1_1_1_1"/>
<font color="Maroon" title="c9">a19</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">a19</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c10">b19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T5" title="LATTICES:th.5">LATTICES:5</a></span>;<br/>
<a NAME="E20:9_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c7">a1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">b1</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E21:9_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">b1</font></span>)</span></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<a NAME="E22:9_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a19</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a19</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c10">b19</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E23:9_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">b1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_1_1_1"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="filter_0.html#T6" title="FILTER_0:th.6">FILTER_0:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:9_1_1_1_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:9_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="ref" href="lattices.html#T8" title="LATTICES:th.8">LATTICES:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E4:9_1_1_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:9_1_1_1_1_2"/>
<font color="Maroon" title="c9">a19</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">b19</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">b19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T6" title="LATTICES:th.6">LATTICES:6</a></span>;<br/>
<a NAME="E2:9_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">b1</font></span>)</span>,<font color="Maroon" title="c8">b1</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E3:9_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">b1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<a NAME="E4:9_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A10</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a19</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">b19</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">k1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font><span class="kw"> such that </span><br/><a NAME="E6:9_1_1_1_1_2"/><span class="lab"><font color="Green" title="E11">A11</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1_1_1_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="lattice4.html#T6" target="_self" title="LATTICE4:th.6">Th9</a></span>;<br/>
<a NAME="E7:9_1_1_1_1_2"/>
<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_1_1_1_1_2"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="ref" href="lattices.html#T6" title="LATTICES:th.6">LATTICES:6</a></span>;<br/>
<a NAME="E8:9_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E9:9_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c11">k1</font>,<font color="Maroon" title="c8">b1</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E7:9_1_1_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<a NAME="E10:9_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A12</font></span>: 
<font color="Maroon" title="c11">k1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E11:9_1_1_1_1_2"/>
<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_1_1_1_1_2"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="ref" href="lattices.html#T6" title="LATTICES:th.6">LATTICES:6</a></span>;<br/>
<a NAME="E12:9_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E13:9_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c11">k1</font>,<font color="Maroon" title="c7">a1</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E7:9_1_1_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<a NAME="E14:9_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">k1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E15:9_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">k1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">a1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:9_1_1_1_1_2"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="ref" href="filter_0.html#T7" title="FILTER_0:th.7">FILTER_0:7</a></span>;<br/>
<a NAME="E16:9_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c11">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">b1</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E17:9_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">b1</font></span>)</span></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<a NAME="E18:9_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">b1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_1_1_1_1_2"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E19:9_1_1_1_1_2"/>
<font color="Maroon" title="c9">a19</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">b19</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">a19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T6" title="LATTICES:th.6">LATTICES:6</a></span>;<br/>
<a NAME="E20:9_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">b1</font></span>)</span>,<font color="Maroon" title="c7">a1</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E21:9_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">b1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<a NAME="E22:9_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a19</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">b19</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E23:9_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_1_1_2"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="ref" href="filter_0.html#T7" title="FILTER_0:th.7">FILTER_0:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:9_1_1_1_1_2"/>
<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:9_1_1_1_1_2"><span class="lab"><font color="Green" title="E13">A13</font></span></a>, <a class="ref" href="lattices.html#T8" title="LATTICES:th.8">LATTICES:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c6">F</font> as    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#D1" target="_self" title="LATTICE4:def.1">Def1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/>

<a NAME="E11:9_1_1_1"/>
( <font color="Maroon" title="c7">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c7">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E6:9_1_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:9_1_1_1"/>
<font color="Maroon" title="c7">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">set </span><font color="Maroon" title="c3">R</font> =  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">S</font> =  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font>;<br/>
<span class="kw">given </span><font color="Maroon" title="c5">f</font> being    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font><span class="kw"> such that </span><a NAME="E3:9_1_1"/><span class="lab"><font color="Green" title="E5">A14</font></span>: 
<font color="Maroon" title="c5">f</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> <a href="filter_1.html#R1" title="FILTER_1:pred.1">are_isomorphic</a> </span><br/>

<a NAME="E4:9_1_1"/><span class="lab"><font color="Green" title="E6">A15</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></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="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font> iff ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font></span>)</span> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font></span>)</span> &amp; <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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b</font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></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="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font> iff ( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font></span>)</span> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font></span>)</span> &amp; <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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font> ) )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font></span>)</span> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font></span>)</span> &amp; <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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></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="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:9_1_1_2_1"/><span class="lab"><font color="Green" title="E7">A16</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></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="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font></span>)</span> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font></span>)</span> &amp; <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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font> )</span><br/><span class="kw">hence </span><a NAME="E2:9_1_1_2_1"/>
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font></span>)</span> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T15" title="RELAT_1:th.15">RELAT_1:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">a9</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c9">b9</font> = <font color="Maroon" title="c7">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a></span>;<br/><a NAME="E4:9_1_1_2_1"/>
<font color="Maroon" title="c8">a9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_2_1"><span class="lab"><font color="Green" title="E7">A16</font></span></a>, <a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/><a NAME="E5:9_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1_1"><span class="lab"><font color="Green" title="E5">A14</font></span></a>, <a class="ref" href="lattice4.html#T5" target="_self" title="LATTICE4:th.5">Th8</a></span>;<br/><span class="kw">hence </span><a NAME="E6:9_1_1_2_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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:9_1_1_2"/><span class="lab"><font color="Green" title="E7">A17</font></span>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font></span>)</span> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E3:9_1_1_2"/><span class="lab"><font color="Green" title="E8">A18</font></span>: 
<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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></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="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">a9</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c9">b9</font> = <font color="Maroon" title="c7">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E7">A17</font></span></a>, <a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a></span>;<br/>
<a NAME="E5:9_1_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1_1_2"><span class="lab"><font color="Green" title="E8">A18</font></span></a>, <a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E6:9_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">a9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1_1"><span class="lab"><font color="Green" title="E5">A14</font></span></a>, <a class="ref" href="lattice4.html#T5" target="_self" title="LATTICE4:th.5">Th8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:9_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></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="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E7">A19</font></span>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> = 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a></span>
;<br/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">f</font> = 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1_1"><span class="lab"><font color="Green" title="E5">A14</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>
<br/>.= 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a></span>
;<br/>
<a NAME="E7:9_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font>, <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1_1"><span class="lab"><font color="Green" title="E5">A14</font></span></a>, <a class="txt" href="#E5:9_1_1"><span class="lab"><font color="Green" title="E7">A19</font></span></a>, <a class="txt" href="#E4:9_1_1"><span class="lab"><font color="Green" title="E6">A15</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<a NAME="E8:9_1_1"/><span class="kw">then </span>
 <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c1">L1</font>, <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <font color="Maroon" title="c2">L2</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#D8" title="WELLORD1:def.8">WELLORD1:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:9_1_1"/>
<font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> <a href="filter_1.html#R1" title="FILTER_1:pred.1">are_isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#D9" title="FILTER_1:def.9">FILTER_1:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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