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

<b>let </b><font color="Maroon">L</font> be  <a href="waybel_1.html#V11">Boolean</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:39"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">L</font> is <a href="lattice3.html#V3">complete</a>
 <b>and </b><br/><a NAME="E2:39"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">X</font> <a href="tarski.html#R1">c=</a>  <a href="waybel15.html#K1" target="_self">ATOM</a> <font color="Maroon">L</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">X</font> )
 ;<br/>

<b>take </b><font color="Maroon">Y</font> =  <a href="waybel15.html#K1" target="_self">ATOM</a> <font color="Maroon">L</font>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">x'</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">x'</font> );<br/>
<a NAME="E3:39"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">Y</font></span>)</span>  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">Y</font></span>)</span>;<br/>

<a NAME="E1:39_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="waybel_8.html#T28">WAYBEL_8:28</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>take </b><font color="Maroon">y</font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">x'</font>;<br/>

<b>take </b>
<font color="Maroon">x'</font>
;<br/>

<b>thus </b><a NAME="E3:39_1"/>
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">x'</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:39"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">Y</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S10">FUNCT_2:sch 10</a>(<a class="txt" href="waybel15.html#E2:39"><i><font color="Green">E34</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">Y</font></span>)</span>, <font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">Y</font></span>)</span>;<br/><b>consider </b><font color="Maroon">z'</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:39_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E3:39_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">z'</font>
 <b>by </b><i><a class="ref" href="waybel15.html#T2" target="_self">Th2</a></i>;<br/><b>consider </b><font color="Maroon">v'</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:39_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">v'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E6:39_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">v'</font>
 <b>by </b><i><a class="ref" href="waybel15.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E7:39_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">z'</font> <a href="tarski.html#R1">c=</a>  <a href="waybel15.html#K1" target="_self">ATOM</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="waybel_8.html#T28">WAYBEL_8:28</a></i>;<br/><a NAME="E8:39_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">v'</font> <a href="tarski.html#R1">c=</a>  <a href="waybel15.html#K1" target="_self">ATOM</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="waybel_8.html#T28">WAYBEL_8:28</a></i>;<br/><b>assume </b><a NAME="E9:39_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 ;<br/><a NAME="E10:39_2"/><b>then </b>
( <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">v</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">z</font> )
 <b>by </b><i>, , , , , , <a class="ref" href="waybel15.html#T3" target="_self">Th3</a>, </i>;<br/><b>hence </b><a NAME="E11:39_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:39"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D1">WAYBEL_1:def 1</a></i>;<br/>
<a NAME="E8:39"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon">K</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:39_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">K</font> <a href="tarski.html#R1">c=</a>  <a href="waybel15.html#K1" target="_self">ATOM</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E4:39_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="txt" href="waybel15.html#E1:39"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E5:39_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">K</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="waybel_8.html#T28">WAYBEL_8:28</a></i>;<br/>
<a NAME="E6:39_3"/><b>then </b>
( <font color="Maroon">K</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">Y</font></span>)</span> &amp; not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> is <a href="xboole_0.html#V1">empty</a> )
 
;<br/>
<a NAME="E7:39_3"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">K</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">K'</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:39_3"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">K'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E10:39_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">K'</font>
 <b>by </b><i><a class="ref" href="waybel15.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>thus </b><a NAME="E11:39_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="waybel15.html#T3" target="_self">Th3</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:39"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">Y</font></span>)</span>, <font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">Y</font></span>)</span>;<br/><b>consider </b><font color="Maroon">z'</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:39_4"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E3:39_4"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">z'</font>
 <b>by </b><i><a class="ref" href="waybel15.html#T2" target="_self">Th2</a></i>;<br/><b>consider </b><font color="Maroon">v'</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:39_4"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">v'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E6:39_4"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">v'</font>
 <b>by </b><i><a class="ref" href="waybel15.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E7:39_4"/><i><font color="Green">E108</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">z'</font>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><a NAME="E8:39_4"/><i><font color="Green">E109</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">v'</font>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><a NAME="E9:39_4"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">z'</font> <a href="tarski.html#R1">c=</a>  <a href="waybel15.html#K1" target="_self">ATOM</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="waybel_8.html#T28">WAYBEL_8:28</a></i>;<br/><a NAME="E10:39_4"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">v'</font> <a href="tarski.html#R1">c=</a>  <a href="waybel15.html#K1" target="_self">ATOM</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="waybel_8.html#T28">WAYBEL_8:28</a></i>;<br/><b>thus </b><a NAME="E11:39_4"/>
( <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">v</font> implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_4_1"/>
<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">v</font>
 ;<br/>

<a NAME="E2:39_4_1"/><b>then </b>
<font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/>
<b>hence </b><a NAME="E3:39_4_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, , , <a class="ref" href="waybel15.html#T3" target="_self">Th3</a>, , , <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E12:39_4"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> implies <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">v</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_4_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_4_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 ;<br/>

<a NAME="E2:39_4_2"/><b>then </b>
<font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">v</font>
 
<b>by </b><i>, , , , <a class="ref" href="waybel15.html#T3" target="_self">Th3</a>, , <a class="txt" href="waybel15.html#E3:39"><i><font color="Green">E45</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E3:39_4_2"/>
<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E11:39"/><b>then </b>
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i>, , <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<a NAME="E12:39"/><b>then </b>
 <a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="waybel_1.html#D8">WAYBEL_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E13:39"/>
<font color="Maroon">L</font>, <a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">Y</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="waybel_1.html#T7">WAYBEL_1:7</a></i>;<br/>


</div>
