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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c2">L'</font> = <font color="Maroon" title="c1">L</font> as  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> ;<br/>
<a NAME="E3:29"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c2">L'</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>
 
;<br/>
<b>consider </b><font color="Maroon" title="c3">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:29"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c3">g</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>
 <b>and </b><br/><a NAME="E6:29"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c3">g</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 <b>and </b><br/><a NAME="E7:29"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c3">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E8:29"/><i><font color="Green" title="E29">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:29"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="waybel13.html#T24" target="_self" title="WAYBEL13:th.24">Th24</a></i>;<br/>
<b>take </b><font color="Maroon" title="c4">X</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">S</font> =  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c3">g</font> as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c4">X</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c5">S</font>
;<br/>

<a NAME="E10:29"/><i><font color="Green" title="E30">A7</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:29"/><i><font color="Green" title="E31">A8</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">g1</font> = <font color="Maroon" title="c3">g</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c5">S</font> <b>by </b><i><a class="txt" href="waybel13.html#E10:29"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>thus </b><a NAME="E1:29_1"/>
( <font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">y</font> implies <font color="Maroon" title="c6">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">y</font>
 ;<br/>

<a NAME="E2:29_1_1"/><b>then </b>
 <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c7">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c8">y</font>
 
<b>by </b><i><a class="ref" href="waybel13.html#T1" target="_self" title="WAYBEL13:th.1">Th1</a></i>;<br/>
<a NAME="E3:29_1_1"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c8">y</font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E8:29"><i><font color="Green" title="E29">A6</font></i></a></i>;<br/>
<a NAME="E4:29_1_1"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E8:29"><i><font color="Green" title="E29">A6</font></i></a></i>;<br/>
<a NAME="E5:29_1_1"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T2" title="YELLOW_1:th.2">YELLOW_1:2</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1_1"/>
<font color="Maroon" title="c6">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:29_1"/>
( <font color="Maroon" title="c6">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font> implies <font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:29_1_2"/><i><font color="Green" title="E32">A9</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="waybel_8.html#V1" title="WAYBEL_8:attr.1">satisfying_axiom_K</a>
 
<b>by </b><i><a class="txt" href="waybel13.html#E1:29"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="waybel_8.html#D4" title="WAYBEL_8:def.4">WAYBEL_8:def 4</a></i>;<br/>
<a NAME="E2:29_1_2"/><i><font color="Green" title="E33">A10</font></i>: 
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="txt" href="waybel13.html#E3:29"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>
<b>assume </b><a NAME="E3:29_1_2"/>
<font color="Maroon" title="c6">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font>
 ;<br/>

<a NAME="E4:29_1_2"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T60" title="YELLOW_0:th.60">YELLOW_0:60</a></i>;<br/>
<a NAME="E5:29_1_2"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T2" title="YELLOW_1:th.2">YELLOW_1:2</a></i>;<br/>
<a NAME="E6:29_1_2"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c8">y</font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E8:29"><i><font color="Green" title="E29">A6</font></i></a></i>;<br/>
<a NAME="E7:29_1_2"/><b>then </b>
 <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c7">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c8">y</font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E8:29"><i><font color="Green" title="E29">A6</font></i></a></i>;<br/>
<a NAME="E8:29_1_2"/><b>then </b>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c8">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel13.html#E2:29_1_2"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="yellow_0.html#T34" title="YELLOW_0:th.34">YELLOW_0:34</a></i>;<br/>
<a NAME="E9:29_1_2"/><b>then </b>
<font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c8">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel13.html#E1:29_1_2"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="waybel_8.html#D3" title="WAYBEL_8:def.3">WAYBEL_8:def 3</a></i>;<br/>
<b>hence </b><a NAME="E10:29_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:29_1_2"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="waybel_8.html#D3" title="WAYBEL_8:def.3">WAYBEL_8:def 3</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E14:29"/><b>then </b><i><font color="Green" title="E32">A11</font></i>: 
<font color="Maroon" title="c6">g1</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 
<b>by </b><i><a class="txt" href="waybel13.html#E7:29"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="waybel13.html#E11:29"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">f1</font> = <font color="Maroon" title="c6">g1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel_0.html#T67" title="WAYBEL_0:th.67">WAYBEL_0:67</a></i>;<br/>
<a NAME="E16:29"/><i><font color="Green" title="E33">A12</font></i>: 
<font color="Maroon" title="c7">f1</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 
<b>by </b><i><a class="txt" href="waybel13.html#E14:29"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="waybel_0.html#T68" title="WAYBEL_0:th.68">WAYBEL_0:68</a></i>;<br/>
<b>thus </b><a NAME="E17:29"/>
<font color="Maroon" title="c5">S</font> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a>
 <b>by </b><i><a class="txt" href="waybel13.html#E3:29"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="waybel13.html#E5:29"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="yellow_2.html#T35" title="YELLOW_2:th.35">YELLOW_2:35</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">Y</font> be  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">S</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:29_2"/><i><font color="Green" title="E34">A13</font></i>: 
<font color="Maroon" title="c8">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E2:29_2"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c8">Y</font>, <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c4">X</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">X2</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">Y</font></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_2_1_1"/>
<font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">g1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">X2</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c11">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_2_1_1"/><i><font color="Green" title="E35">A14</font></i>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 <b>and </b><br/><a NAME="E4:29_2_1_1"/><i><font color="Green" title="E36">A15</font></i>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X2</font>
 <b>and </b><br/><a NAME="E5:29_2_1_1"/><i><font color="Green" title="E37">A16</font></i>: 
<font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">u</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:29_2_1_1"/>
<font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">g1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">Y</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E3:29_2_1_1"><i><font color="Green" title="E35">A14</font></i></a>, <a class="txt" href="waybel13.html#E4:29_2_1_1"><i><font color="Green" title="E36">A15</font></i></a>, <a class="txt" href="waybel13.html#E5:29_2_1_1"><i><font color="Green" title="E37">A16</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:29_2_1_1"/><b>then </b>
<font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">g1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">Y</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E7:29"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="funct_1.html#T155" title="FUNCT_1:th.155">FUNCT_1:155</a></i>;<br/><b>hence </b><a NAME="E8:29_2_1_1"/>
<font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E11:29"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="funct_1.html#T147" title="FUNCT_1:th.147">FUNCT_1:147</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c10">Y1</font> = <font color="Maroon" title="c6">g1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">X2</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font><b> such that </b><br/><a NAME="E4:29_2_1"/><i><font color="Green" title="E35">A17</font></i>: 
<font color="Maroon" title="c11">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 <b>and </b><br/><a NAME="E5:29_2_1"/><i><font color="Green" title="E36">A18</font></i>: 
<font color="Maroon" title="c11">y1</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c10">Y1</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:29_2"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="waybel_0.html#T1" title="WAYBEL_0:th.1">WAYBEL_0:1</a></i>;<br/><b>take </b><font color="Maroon" title="c12">f1y1</font> = <font color="Maroon" title="c7">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y1</font>;<br/><a NAME="E6:29_2_1"/>
( not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c11">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S</font> )
 ;<br/><a NAME="E7:29_2_1"/><b>then </b>
<font color="Maroon" title="c11">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E8:29_2_1"/>
<font color="Maroon" title="c12">f1y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">Y</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E4:29_2_1"><i><font color="Green" title="E35">A17</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E9:29_2_1"/>
( <font color="Maroon" title="c9">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp; not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E10:29_2_1"/><b>then </b>
<font color="Maroon" title="c9">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E11:29_2_1"/><b>then </b><i><font color="Green" title="E37">A19</font></i>: 
<font color="Maroon" title="c9">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">X2</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T146" title="FUNCT_1:th.146">FUNCT_1:146</a></i>;<br/><a NAME="E12:29_2_1"/><i><font color="Green" title="E38">A20</font></i>: 
<font color="Maroon" title="c6">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">X2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">X2</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E7:29"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="funct_1.html#T152" title="FUNCT_1:th.152">FUNCT_1:152</a></i>;<br/><font color="Maroon" title="c7">f1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Y1</font> = 
<font color="Maroon" title="c6">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">X2</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel13.html#E7:29"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="funct_1.html#T155" title="FUNCT_1:th.155">FUNCT_1:155</a></i>
<br/>.= 
<font color="Maroon" title="c9">X2</font>
<b>by </b><i><a class="txt" href="waybel13.html#E11:29_2_1"><i><font color="Green" title="E37">A19</font></i></a>, <a class="txt" href="waybel13.html#E12:29_2_1"><i><font color="Green" title="E38">A20</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>
;<br/><b>hence </b><a NAME="E14:29_2_1"/>
<font color="Maroon" title="c12">f1y1</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c9">X2</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E16:29"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="waybel13.html#E5:29_2_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="ref" href="waybel13.html#T19" target="_self" title="WAYBEL13:th.19">Th19</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c9">X1</font> = <font color="Maroon" title="c7">f1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">Y</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel_0.html#T1" title="WAYBEL_0:th.1">WAYBEL_0:1</a></i>;<br/><i><font color="Green" title="E35">A21</font></i>: <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">X1</font> = 
<font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">Y</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel13.html#E7:29"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="funct_1.html#T155" title="FUNCT_1:th.155">FUNCT_1:155</a></i>
<br/>.= 
<font color="Maroon" title="c8">Y</font>
<b>by </b><i><a class="txt" href="waybel13.html#E11:29"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="funct_1.html#T147" title="FUNCT_1:th.147">FUNCT_1:147</a></i>
;<br/><a NAME="E6:29_2"/><i><font color="Green" title="E36">A22</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c9">X1</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E3:29"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/><a NAME="E7:29_2"/><i><font color="Green" title="E37">A23</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c9">X1</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E6:29"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></i>;<br/><a NAME="E8:29_2"/>
(  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c9">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp; not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c4">X</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 ;<br/><a NAME="E9:29_2"/><b>then </b>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c9">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E10:29_2"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c9">X1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">g</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E11:29_2"/><b>then </b>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">X1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">g</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E6:29_2"><i><font color="Green" title="E36">A22</font></i></a>, <a class="txt" href="waybel13.html#E7:29_2"><i><font color="Green" title="E37">A23</font></i></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></i>;<br/><b>hence </b><a NAME="E12:29_2"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c8">Y</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E5:29_2"><i><font color="Green" title="E35">A21</font></i></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E19:29"/>
<font color="Maroon" title="c5">S</font> is <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D4" title="WAYBEL_0:def.4">WAYBEL_0:def 4</a></i>;<br/>

<b>thus </b><a NAME="E20:29"/>
<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c5">S</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="waybel13.html#E14:29"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a></i>;<br/>


</div>
