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

<b>thus </b><a NAME="E1:11_1_1"/>
( <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="filter_1.html#R1">are_isomorphic</a>  implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="lattice4.html#M1" target="_self">Homomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="lattice4.html#V3" target="_self">isomorphism</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="filter_1.html#R1">are_isomorphic</a> 
 ;<br/>

<a NAME="E2:11_1_1_1"/><b>then </b>
 <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font>, <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#R4">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="filter_1.html#D9">FILTER_1:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:11_1_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#R3">is_isomorphism_of</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font>, <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="wellord1.html#D8">WELLORD1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E5:11_1_1_1"/><i><font color="Green">E9</font></i>: 
(  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="filter_1.html#T33">FILTER_1:33</a></i>;<br/>
<a NAME="E6:11_1_1_1"/><i><font color="Green">E10</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font> ) ) ) )
 
<b>by </b><i><a class="txt" href="lattice4.html#E4:11_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="lattice4.html#E5:11_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E2:11_1_1_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="lattice4.html#V2" target="_self">epimorphism</a>
 <b>by </b><i><a class="txt" href="lattice4.html#E5:11_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="lattice4.html#E6:11_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="lattice4.html#D3" target="_self">Def3</a></i>;<br/><b>thus </b><a NAME="E3:11_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:11_1_1_1_1_1"/><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="lattice4.html#E2:11_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="lattice4.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E3:11_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice4.html#E2:11_1_1_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="lattices.html#T22">LATTICES:22</a></i>;<br/>
<a NAME="E4:11_1_1_1_1_1"/><b>then </b>
( <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E5:11_1_1_1_1_1"/><b>then </b>
( <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice4.html#E4:11_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="lattice4.html#E5:11_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E6:11_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E7:11_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="filter_0.html#T6">FILTER_0:6</a></i>;<br/>
<a NAME="E8:11_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E9:11_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="lattice4.html#E4:11_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E10:11_1_1_1_1_1"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice4.html#E2:11_1_1_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E11:11_1_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="lattices.html#T22">LATTICES:22</a></i>;<br/>
<a NAME="E12:11_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E13:11_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="lattice4.html#E4:11_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E14:11_1_1_1_1_1"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E15:11_1_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="lattices.html#T22">LATTICES:22</a></i>;<br/>
<a NAME="E16:11_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E17:11_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="lattice4.html#E4:11_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E18:11_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E19:11_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice4.html#E14:11_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="filter_0.html#T6">FILTER_0:6</a></i>;<br/>
<b>hence </b><a NAME="E20:11_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice4.html#E10:11_1_1_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:11_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:11_1_1_1_1_2"/><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="lattice4.html#E2:11_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="lattice4.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E3:11_1_1_1_1_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice4.html#E2:11_1_1_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="lattices.html#T23">LATTICES:23</a></i>;<br/>
<a NAME="E4:11_1_1_1_1_2"/><b>then </b>
( <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E5:11_1_1_1_1_2"/><b>then </b>
( <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice4.html#E4:11_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="lattice4.html#E5:11_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E6:11_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E7:11_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="filter_0.html#T7">FILTER_0:7</a></i>;<br/>
<a NAME="E8:11_1_1_1_1_2"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E9:11_1_1_1_1_2"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="lattice4.html#E4:11_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E10:11_1_1_1_1_2"/><b>then </b><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice4.html#E2:11_1_1_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E11:11_1_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>10</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="lattices.html#T23">LATTICES:23</a></i>;<br/>
<a NAME="E12:11_1_1_1_1_2"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E13:11_1_1_1_1_2"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="lattice4.html#E4:11_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E14:11_1_1_1_1_2"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E15:11_1_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>10</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="lattices.html#T23">LATTICES:23</a></i>;<br/>
<a NAME="E16:11_1_1_1_1_2"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E17:11_1_1_1_1_2"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="lattice4.html#E4:11_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E18:11_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E19:11_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice4.html#E14:11_1_1_1_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="filter_0.html#T7">FILTER_0:7</a></i>;<br/>
<b>hence </b><a NAME="E20:11_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice4.html#E10:11_1_1_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="lattice4.html#M1" target="_self">Homomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="lattice4.html#D1" target="_self">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<a NAME="E10:11_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> is <a href="lattice4.html#V1" target="_self">monomorphism</a> &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="lattice4.html#V2" target="_self">epimorphism</a> )
 
<b>by </b><i><a class="txt" href="lattice4.html#E5:11_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="lattice4.html#E6:11_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="lattice4.html#D2" target="_self">Def2</a>, <a class="ref" href="lattice4.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E11:11_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="lattice4.html#V3" target="_self">isomorphism</a>
 <b>by </b><i><a class="ref" href="lattice4.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>3</sub></font> being    <a href="lattice4.html#M1" target="_self">Homomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><a NAME="E2:11_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="lattice4.html#V3" target="_self">isomorphism</a>
 ;<br/>

<a NAME="E3:11_1_1"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="lattice4.html#V1" target="_self">monomorphism</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="lattice4.html#V2" target="_self">epimorphism</a> )
 
<b>by </b><i><a class="txt" href="lattice4.html#E2:11_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="lattice4.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E4:11_1_1"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="lattice4.html#D2" target="_self">Def2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<i><font color="Green">E11</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="filter_1.html#T33">FILTER_1:33</a></i>
;<br/>
<i><font color="Green">E12</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="lattice4.html#E3:11_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="lattice4.html#D3" target="_self">Def3</a></i>
<br/>.= 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="filter_1.html#T33">FILTER_1:33</a></i>
;<br/>
<a NAME="E7:11_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:11_1_1_4_1"/><i><font color="Green">E13</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>hence </b><a NAME="E2:11_1_1_4_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="filter_1.html#T33">FILTER_1:33</a></i>;<br/><a NAME="E4:11_1_1_4_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="lattice4.html#E1:11_1_1_4_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/><a NAME="E5:11_1_1_4_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="lattice4.html#E3:11_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="lattice4.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E6:11_1_1_4_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:11_1_1_4"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="filter_1.html#T33">FILTER_1:33</a></i>;<br/>
<a NAME="E4:11_1_1_4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="lattice4.html#E2:11_1_1_4"><i><font color="Green">E13</font></i></a>, <a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E5:11_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="lattice4.html#E3:11_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="lattice4.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E6:11_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:11_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#R3">is_isomorphism_of</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font>, <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="lattice4.html#E4:11_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="lattice4.html#E5:11_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="lattice4.html#E6:11_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E9:11_1_1"/><b>then </b>
 <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font>, <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#R4">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="wellord1.html#D8">WELLORD1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E10:11_1_1"/>
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="filter_1.html#R1">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="filter_1.html#D9">FILTER_1:def 9</a></i>;<br/>


</div>
