<?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="E1:60"/><i><font color="Green" title="E25">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#K3" title="WAYBEL_3:func.3">.</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="E2:60"/><b>then </b><i><font color="Green" title="E26">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#K3" title="WAYBEL_3:func.3">.</a> <font color="Olive" title="b1">i</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>
 
;<br/>
<a NAME="E3:60"/>
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#K3" title="WAYBEL_3:func.3">.</a> <font color="Olive" title="b1">i</font> is <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E1:60"><i><font color="Green" title="E25">A1</font></i></a></i>;<br/>
<b>then 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#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a> <b>by </b><i><a class="txt" href="waybel_3.html#E2:60"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="waybel_3.html#T29" target="_self" title="WAYBEL_3:th.29">Th29</a>, <a class="ref" href="waybel_3.html#T30" target="_self" title="WAYBEL_3:th.30">Th30</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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/><b>deffunc </b><font color="Maroon">H<sub>1</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#K3" title="WAYBEL_3:func.3">.</a> $1</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> <font color="Maroon" title="c4">X</font>,$1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c5">f</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E2:60_1"/><i><font color="Green" title="E27">A3</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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></i><a NAME="E3:60_1"/><i><font color="Green" title="E28">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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="E1:60_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_3.html#E2:60_1"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E2:60_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</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#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c5">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="waybel_3.html#E3:60_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="waybel_3.html#T27" target="_self" title="WAYBEL_3:th.27">Th27</a></i>;<br/><a NAME="E6:60_1"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c4">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:60_1_2"/><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">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="E1:60_1_2_1"/>
( <font color="Maroon" title="c7">x</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</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> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c8">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> &amp; <font color="Maroon" title="c6">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">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_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c8">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E1:60"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="waybel_3.html#E2:60_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="waybel_3.html#E1:60_1_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>;<br/><b>hence </b><a NAME="E2:60_1_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T24" title="YELLOW_2:th.24">YELLOW_2:24</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:60_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T28" target="_self" title="WAYBEL_3:th.28">Th28</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">g</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/><b>assume </b><a NAME="E1:60_1_3"/><i><font color="Green" title="E30">A7</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c7">g</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_3_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:60_1_3_1"/>
<font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font>
 ;<br/><b>let </b><font color="Maroon" title="c8">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="E2:60_1_3_1"/><i><font color="Green" title="E31">A8</font></i>: 
( <font color="Maroon" title="c6">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">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_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c8">i</font></span>)</span> &amp; <font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c8">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> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E1:60"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="waybel_3.html#E2:60_1"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/><a NAME="E3:60_1_3_1"/>
<font color="Maroon" title="c7">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c8">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">a</font> be   <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#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c8">i</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:60_1_3_1_1"/>
<font color="Maroon" title="c9">a</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> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c8">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:60_1_3_1_1"/><i><font color="Green" title="E32">A9</font></i>: 
( <font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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/>
<b>reconsider </b><font color="Maroon" title="c11">h</font> = <font color="Maroon" title="c10">h</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="waybel_3.html#E3:60_1_3_1_1"><i><font color="Green" title="E32">A9</font></i></a></i>;<br/>
<a NAME="E5:60_1_3_1_1"/>
<font color="Maroon" title="c11">h</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">g</font>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E1:60_1_3"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="waybel_3.html#E3:60_1_3_1_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E6:60_1_3_1_1"/>
<font color="Maroon" title="c9">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E3:60_1_3_1_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="waybel_3.html#T28" target="_self" title="WAYBEL_3:th.28">Th28</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:60_1_3_1"/>
<font color="Maroon" title="c6">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E2:60_1_3_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:60_1_3"/>
<font color="Maroon" title="c6">f</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">g</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T28" target="_self" title="WAYBEL_3:th.28">Th28</a></i>;<br/></div><b>end;</b></div><a NAME="E8:60_1"/><b>then </b>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E6:60_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="yellow_0.html#T15" title="YELLOW_0:th.15">YELLOW_0:15</a></i>;<br/><b>hence </b><a NAME="E9:60_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">X</font>, <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E6:60"/><b>then </b>
<font color="Maroon" title="c3">L</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T53" title="YELLOW_0:th.53">YELLOW_0:53</a></i>;<br/>
<b>hence </b><a NAME="E7:60"/>
 <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</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/>


</div>
