<?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="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a> <a href="yellow16.html#V1" title="YELLOW16:attr.1">Poset-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/><b>let </b><font color="Maroon" title="c3">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/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:44_1"/><i><font color="Green" title="E30">A1</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c3">X</font>, <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font>
 ;<br/><b>let </b><font color="Maroon" title="c4">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/><b>set </b><font color="Maroon" title="c5">f</font> =  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">X</font>;<br/><a NAME="E2:44_1"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">X</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="yellow16.html#E1:44_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></i>;<br/><a NAME="E3:44_1"/><b>then </b><i><font color="Green" title="E31">A2</font></i>: 
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c4">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="c3">X</font>,<font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="yellow16.html#T31" target="_self" title="YELLOW16:th.31">Th31</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</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="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c4">i</font></span>)</span>;<br/><b>assume </b><a NAME="E1:44_1_1"/><i><font color="Green" title="E32">A3</font></i>: 
 <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">i</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c6">x</font>
 ;<br/><b>set </b><font color="Maroon" title="c7">g</font> = <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">x</font>;<br/><a NAME="E2:44_1_1"/><i><font color="Green" title="E33">A4</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">X</font></span>)</span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">X</font></span>)</span> <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="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a>, <a class="ref" href="waybel_3.html#T27" title="WAYBEL_3:th.27">WAYBEL_3:27</a></i>;<br/><a NAME="E3:44_1_1"/><b>then </b><i><font color="Green" title="E34">A5</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">j</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="E1:44_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> <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> <font color="Maroon" title="c3">X</font></span>)</span> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c8">j</font> or ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> ) )
 <b>by </b><i><a class="txt" href="yellow16.html#E3:44_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E2:44_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</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="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c8">j</font></span>)</span>
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c8">g</font> = <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">x</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="yellow16.html#E2:44_1_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="waybel_3.html#T27" title="WAYBEL_3:th.27">WAYBEL_3:27</a></i>;<br/><a NAME="E6:44_1_1"/><i><font color="Green" title="E35">A6</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="yellow16.html#E1:44_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></i>;<br/><a NAME="E7:44_1_1"/>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c8">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">h</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="E1:44_1_1_2"/>
<font color="Maroon" title="c9">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E2:44_1_1_2"/><b>then </b><i><font color="Green" title="E36">A7</font></i>: 
( <font color="Maroon" title="c9">h</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> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c9">h</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c4">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="c3">X</font>,<font color="Maroon" title="c4">i</font> )
 
<b>by </b><i><a class="txt" href="yellow16.html#E6:44_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">j</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="E1:44_1_1_2_1"/>
( <font color="Maroon" title="c8">g</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">j</font> <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> <font color="Maroon" title="c3">X</font></span>)</span> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">j</font> or ( <font color="Maroon" title="c8">g</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> ) )
 <b>by </b><i><a class="txt" href="yellow16.html#E3:44_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E2:44_1_1_2_1"/>
<font color="Maroon" title="c9">h</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">j</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">g</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c10">j</font>
 <b>by </b><i><a class="txt" href="yellow16.html#E1:44_1_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="yellow16.html#E2:44_1_1_2"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a>, <a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:44_1_1_2"/>
<font color="Maroon" title="c9">h</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">g</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:44_1_1"/><b>then </b>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">X</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">g</font>
 <b>by </b><i><a class="txt" href="yellow16.html#E1:44_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></i>;<br/><b>hence </b><a NAME="E9:44_1_1"/>
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Maroon" title="c4">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="yellow16.html#E3:44_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:44_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="yellow16.html#E3:44_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:44"/>
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#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c3">X</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Olive" title="b1">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">f</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="E4:44"/>
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="c4">f</font> <a href="waybel_3.html#K7" title="WAYBEL_3:func.7">.</a> <font color="Olive" title="b1">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="c3">X</font>,<font color="Olive" title="b1">i</font></span>)</span>
 <b>and </b><br/><a NAME="E5:44"/><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font>
 <b>and </b><br/><a NAME="E6:44"/><i><font color="Green" title="E31">A9</font></i>: 
for <font color="Olive" title="b1">g</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>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">g</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">g</font>
 <b>by </b><i><a class="txt" href="yellow16.html#E37"><i><font color="Green" title="E25">Lm2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:44"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c3">X</font>, <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font>
 <b>by </b><i><a class="txt" href="yellow16.html#E5:44"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="yellow16.html#E6:44"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></i>;<br/>


</div>
