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

<b>let </b><font color="Maroon" title="c1">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">J</font> be  <a href="yellow_1.html#V1" title="YELLOW_1:attr.1">RelStr-yielding</a> <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a> <a href="waybel_3.html#V5" title="WAYBEL_3:attr.5">reflexive-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/>



<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E23">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> <font color="Olive" title="b1">i</font> is  <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="E2:28"/><b>then </b><i><font color="Green" title="E24">A2</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> <font color="Olive" title="b1">i</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 
;<br/>
<a NAME="E3:28"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> <font color="Olive" title="b1">i</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">antisymmetric</a>
 
<b>by </b><i><a class="txt" href="waybel13.html#E1:28"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>
<a NAME="E4:28"/><b>then </b><i><font color="Green" title="E25">A3</font></i>: 
 <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">antisymmetric</a>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T30" title="WAYBEL_3:th.30">WAYBEL_3:30</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">L</font> =  <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font> as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <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> <b>by </b><i><a class="txt" href="waybel13.html#E2:28"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="waybel_3.html#T31" title="WAYBEL_3:th.31">WAYBEL_3:31</a></i>;<br/>
<a NAME="E6:28"/><i><font color="Green" title="E26">A4</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="c3">L</font> holds <br/> ( not  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Olive" title="b1">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Olive" title="b1">x</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font>;<br/><b>assume </b><a NAME="E1:28_1_1"/>
( <font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">x</font> )
 ;<br/><a NAME="E2:28_1_1"/><b>then </b><i><font color="Green" title="E27">A5</font></i>: 
( <font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c5">c</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> &amp; <font color="Maroon" title="c6">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c6">d</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> )
 <b>by </b><i><a class="ref" href="waybel_8.html#T4" title="WAYBEL_8:th.4">WAYBEL_8:4</a></i>;<br/><b>take </b><font color="Maroon" title="c7">e</font> = <font color="Maroon" title="c5">c</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c6">d</font>;<br/><a NAME="E3:28_1_1"/><i><font color="Green" title="E28">A6</font></i>: 
( <font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">c</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c6">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">c</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c6">d</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><a NAME="E4:28_1_1"/>
<font color="Maroon" title="c5">c</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">c</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E2:28_1_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="waybel_3.html#D2" title="WAYBEL_3:def.2">WAYBEL_3:def 2</a></i>;<br/><a NAME="E5:28_1_1"/><b>then </b><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">c</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E3:28_1_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></i>;<br/><a NAME="E6:28_1_1"/>
<font color="Maroon" title="c6">d</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E2:28_1_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="waybel_3.html#D2" title="WAYBEL_3:def.2">WAYBEL_3:def 2</a></i>;<br/><a NAME="E7:28_1_1"/><b>then </b>
<font color="Maroon" title="c6">d</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">c</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E3:28_1_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></i>;<br/><a NAME="E8:28_1_1"/><b>then </b>
<font color="Maroon" title="c5">c</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c6">d</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">c</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E5:28_1_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="waybel_3.html#T3" title="WAYBEL_3:th.3">WAYBEL_3:3</a></i>;<br/><a NAME="E9:28_1_1"/><b>then </b><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c6">d</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 <b>by </b><i><a class="ref" href="waybel_3.html#D2" title="WAYBEL_3:def.2">WAYBEL_3:def 2</a></i>;<br/><a NAME="E10:28_1_1"/>
<font color="Maroon" title="c5">c</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c6">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E2:28_1_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><b>hence </b><a NAME="E11:28_1_1"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E9:28_1_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="waybel_8.html#T4" title="WAYBEL_8:th.4">WAYBEL_8:4</a></i>;<br/><b>thus </b><a NAME="E12:28_1_1"/>
( <font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">e</font> &amp; <font color="Maroon" title="c6">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">e</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:28_1"/>
( not  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">x</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:28"/><i><font color="Green" title="E27">A9</font></i>: 
<font color="Maroon" title="c3">L</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>
 
<b>by </b><i><a class="txt" href="waybel13.html#E2:28"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="waybel_3.html#T31" title="WAYBEL_3:th.31">WAYBEL_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font></span>)</span>;<br/><a NAME="E1:28_2"/><i><font color="Green" title="E28">A10</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds   <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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>;<br/>

<a NAME="E1:28_2_1"/><i><font color="Green" title="E29">A11</font></i>: 
 <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c5">i</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_2_1_1"/>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c5">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:28_2_1_1"/><i><font color="Green" title="E30">A12</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">x</font>
 <b>and </b><br/><a NAME="E4:28_2_1_1"/><i><font color="Green" title="E31">A13</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>;<br/>
<a NAME="E5:28_2_1_1"/>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font></span>)</span> : ( <font color="Maroon" title="c4">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">y</font> &amp; <font color="Olive" title="b1">y</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel13.html#E3:28_2_1_1"><i><font color="Green" title="E30">A12</font></i></a>, <a class="ref" href="waybel_8.html#D2" title="WAYBEL_8:def.2">WAYBEL_8:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">f1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font></span>)</span><b> such that </b><br/><a NAME="E7:28_2_1_1"/><i><font color="Green" title="E32">A14</font></i>: 
<font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font>
 <b>and </b><br/><a NAME="E8:28_2_1_1"/><i><font color="Green" title="E33">A15</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">f1</font>
 <b>and </b><br/><a NAME="E9:28_2_1_1"/><i><font color="Green" title="E34">A16</font></i>: 
<font color="Maroon" title="c8">f1</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 ;<br/>
<a NAME="E10:28_2_1_1"/>
<font color="Maroon" title="c8">f1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c8">f1</font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E9:28_2_1_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="ref" href="waybel_3.html#D2" title="WAYBEL_3:def.2">WAYBEL_3:def 2</a></i>;<br/>
<a NAME="E11:28_2_1_1"/><b>then </b>
<font color="Maroon" title="c8">f1</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c8">f1</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E2:28"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="waybel_3.html#T33" title="WAYBEL_3:th.33">WAYBEL_3:33</a></i>;<br/>
<a NAME="E12:28_2_1_1"/><b>then </b><i><font color="Green" title="E35">A17</font></i>: 
<font color="Maroon" title="c8">f1</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D2" title="WAYBEL_3:def.2">WAYBEL_3:def 2</a></i>;<br/>
<a NAME="E13:28_2_1_1"/>
<font color="Maroon" title="c8">f1</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E8:28_2_1_1"><i><font color="Green" title="E33">A15</font></i></a>, <a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></i>;<br/>
<b>hence </b><a NAME="E14:28_2_1_1"/>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E4:28_2_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="waybel13.html#E7:28_2_1_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="waybel13.html#E12:28_2_1_1"><i><font color="Green" title="E35">A17</font></i></a>, <a class="ref" href="waybel_8.html#T4" title="WAYBEL_8:th.4">WAYBEL_8:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_2_1"/>
 <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c5">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_2_1_2"/>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 ;<br/>

<a NAME="E2:28_2_1_2"/><b>then </b>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> <font color="Maroon" title="c5">i</font></span>)</span> : ( <font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">y</font> &amp; <font color="Olive" title="b1">y</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_8.html#D2" title="WAYBEL_8:def.2">WAYBEL_8:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">v1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> <font color="Maroon" title="c5">i</font></span>)</span><b> such that </b><br/><a NAME="E4:28_2_1_2"/><i><font color="Green" title="E30">A18</font></i>: 
<font color="Maroon" title="c7">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v</font>
 <b>and </b><br/><a NAME="E5:28_2_1_2"/><i><font color="Green" title="E31">A19</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">v1</font>
 <b>and </b><br/><a NAME="E6:28_2_1_2"/><i><font color="Green" title="E32">A20</font></i>: 
<font color="Maroon" title="c7">v1</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> <font color="Maroon" title="c5">i</font></span>)</span> = <font color="Maroon" title="c7">v1</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> $1</span>)</span> =  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c8">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:28_2_1_2"/><i><font color="Green" title="E33">A21</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>and </b><br/><a NAME="E9:28_2_1_2"/><i><font color="Green" title="E34">A22</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] implies <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] implies <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">j</font>) ) )
 <b>from </b><i><a class="ref" href="partfun1.html#S4" title="PARTFUN1:sch.4">PARTFUN1:sch 4</a>();<br/></i>
<a NAME="E10:28_2_1_2"/><i><font color="Green" title="E35">A23</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v1</font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E9:28_2_1_2"><i><font color="Green" title="E34">A22</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2_1_2_1_1_1"/>
( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> or <font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:28_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:28_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E9:28_2_1_2"><i><font color="Green" title="E34">A22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:28_2_1_2_1_1_1_2"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font>
 ;<br/><div class="add"><a NAME="E2:28_2_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E9:28_2_1_2"><i><font color="Green" title="E34">A22</font></i></a></i>;<br/><b>hence </b><a NAME="E3:28_2_1_2_1_1_1_2"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:28_2_1_2_1"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">f</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font></span>)</span> <b>by </b><i><a class="txt" href="waybel13.html#E8:28_2_1_2"><i><font color="Green" title="E33">A21</font></i></a>, <a class="ref" href="waybel_3.html#T27" title="WAYBEL_3:th.27">WAYBEL_3:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>;<br/><a NAME="E1:28_2_1_2_2"/><i><font color="Green" title="E36">A24</font></i>: 
<font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> <font color="Maroon" title="c10">k</font> is  <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>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:28"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2_1_2_2_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> or <font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:28_2_1_2_2_1_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:28_2_1_2_2_1_1_1"/>
<font color="Maroon" title="c9">f</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E5:28_2_1_2"><i><font color="Green" title="E31">A19</font></i></a>, <a class="txt" href="waybel13.html#E9:28_2_1_2"><i><font color="Green" title="E34">A22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:28_2_1_2_2_1_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font>
 ;<br/><div class="add"><a NAME="E2:28_2_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">f</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E9:28_2_1_2"><i><font color="Green" title="E34">A22</font></i></a></i>;<br/><b>hence </b><a NAME="E3:28_2_1_2_2_1_1_2"/>
<font color="Maroon" title="c9">f</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:28_2_1_2_2"><i><font color="Green" title="E36">A24</font></i></a>, <a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:28_2_1_2_2"/>
<font color="Maroon" title="c9">f</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">k</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E14:28_2_1_2"/><b>then </b><i><font color="Green" title="E36">A25</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></i>;<br/>
<a NAME="E15:28_2_1_2"/><i><font color="Green" title="E37">A26</font></i>: 
 ex <font color="Olive" title="b1">K</font> being <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="c1">I</font> st <br/>for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>  st not <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> holds <br/><font color="Maroon" title="c9">f</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_2_3"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c10">K</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>

<b>let </b><font color="Maroon" title="c11">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>;<br/>

<b>assume </b><a NAME="E1:28_2_1_2_3"/>
not <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">K</font>
 ;<br/>

<a NAME="E2:28_2_1_2_3"/><b>then </b>
<font color="Maroon" title="c11">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:28_2_1_2_3"/>
<font color="Maroon" title="c9">f</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> <font color="Maroon" title="c11">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E9:28_2_1_2"><i><font color="Green" title="E34">A22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>;<br/><a NAME="E1:28_2_1_2_4"/><i><font color="Green" title="E38">A27</font></i>: 
<font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> <font color="Maroon" title="c10">k</font> is  <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>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:28"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2_1_2_4_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> or <font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:28_2_1_2_4_1_1_1"/><i><font color="Green" title="E39">A28</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font>
 ;<br/><div class="add"><a NAME="E2:28_2_1_2_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">f</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v1</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E9:28_2_1_2"><i><font color="Green" title="E34">A22</font></i></a></i>;<br/><b>hence </b><a NAME="E3:28_2_1_2_4_1_1_1"/>
<font color="Maroon" title="c9">f</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">k</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c9">f</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E6:28_2_1_2"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="waybel13.html#E1:28_2_1_2_4_1_1_1"><i><font color="Green" title="E39">A28</font></i></a>, <a class="ref" href="waybel_3.html#D2" title="WAYBEL_3:def.2">WAYBEL_3:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:28_2_1_2_4_1_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font>
 ;<br/><div class="add"><a NAME="E2:28_2_1_2_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">f</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E9:28_2_1_2"><i><font color="Green" title="E34">A22</font></i></a></i>;<br/><a NAME="E3:28_2_1_2_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">f</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">k</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:28_2_1_2_4"><i><font color="Green" title="E38">A27</font></i></a>, <a class="ref" href="waybel_3.html#T15" title="WAYBEL_3:th.15">WAYBEL_3:15</a></i>;<br/><b>hence </b><a NAME="E4:28_2_1_2_4_1_1_2"/>
<font color="Maroon" title="c9">f</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">k</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c9">f</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#D2" title="WAYBEL_3:def.2">WAYBEL_3:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:28_2_1_2_4"/>
<font color="Maroon" title="c9">f</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">k</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c9">f</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">k</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E17:28_2_1_2"/><b>then </b>
<font color="Maroon" title="c9">f</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c9">f</font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E2:28"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="waybel13.html#E15:28_2_1_2"><i><font color="Green" title="E37">A26</font></i></a>, <a class="ref" href="waybel_3.html#T33" title="WAYBEL_3:th.33">WAYBEL_3:33</a></i>;<br/>
<a NAME="E18:28_2_1_2"/><b>then </b>
<font color="Maroon" title="c9">f</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D2" title="WAYBEL_3:def.2">WAYBEL_3:def 2</a></i>;<br/>
<a NAME="E19:28_2_1_2"/><b>then </b>
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E14:28_2_1_2"><i><font color="Green" title="E36">A25</font></i></a>, <a class="ref" href="waybel_8.html#T4" title="WAYBEL_8:th.4">WAYBEL_8:4</a></i>;<br/>
<b>hence </b><a NAME="E20:28_2_1_2"/>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E4:28_2_1_2"><i><font color="Green" title="E30">A18</font></i></a>, <a class="txt" href="waybel13.html#E10:28_2_1_2"><i><font color="Green" title="E35">A23</font></i></a>, <a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span> = 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel13.html#E1:28_2_1"><i><font color="Green" title="E29">A11</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="txt" href="waybel13.html#E2:28"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="waybel_3.html#T32" title="WAYBEL_3:th.32">WAYBEL_3:32</a></i>
;<br/><br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>;<br/><a NAME="E1:28_2_2"/>
<font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> <font color="Maroon" title="c5">i</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#E1:28"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/><a NAME="E2:28_2_2"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
 <b>by </b><i><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="E3:28_2_2"/>
<font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:28_2"><i><font color="Green" title="E28">A10</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:28_2"/><b>then </b><i><font color="Green" title="E29">A29</font></i>: 
<font color="Maroon" title="c4">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="c4">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>;<br/><a NAME="E1:28_2_3"/>
<font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K6" title="WAYBEL_3:func.6">.</a> <font color="Maroon" title="c5">i</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#E1:28"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/><a NAME="E2:28_2_3"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
 <b>by </b><i><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="E3:28_2_3"/>
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:28_2"><i><font color="Green" title="E28">A10</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:28_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="c4">x</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></i>;<br/><b>hence </b><a NAME="E6:28_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E4:28"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="waybel13.html#E3:28_2"><i><font color="Green" title="E29">A29</font></i></a>, <a class="ref" href="yellow_0.html#D3" title="YELLOW_0:def.3">YELLOW_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:28"/><b>then </b>
 <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font> is <a href="waybel_8.html#V1" title="WAYBEL_8:attr.1">satisfying_axiom_K</a>
 
<b>by </b><i><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="E10:28"/>
 <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font> is  <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>
 <b>by </b><i><a class="txt" href="waybel13.html#E6:28"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="waybel13.html#E7:28"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="waybel_8.html#D4" title="WAYBEL_8:def.4">WAYBEL_8:def 4</a></i>;<br/>


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