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

<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">J</font> be  <a href="yellow_1.html#V1">RelStr-yielding</a> <a href="waybel_3.html#V4" target="_self">non-Empty</a> <a href="waybel_3.html#V5" target="_self">reflexive-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E1:61"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">J</font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Olive">i</font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">L</font> =  <a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font> as  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i/>;<br/>
<a NAME="E3:61"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">L</font> is <a href="lattice3.html#V3">complete</a>
 
;<br/>
<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/>



<a NAME="E4:61"/><i><font color="Green">E39</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<a NAME="E5:61"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">i</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">i</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:61_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:61_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E5:61_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E6:61_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:61"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">J</font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">i</font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">i</font></span>)</span>;<br/><b>assume </b><a NAME="E1:61_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">a</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font>
 ;<br/><b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font>;<br/><a NAME="E2:61_2"/><i><font color="Green">E48</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 <b>by </b><i>, <a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><a NAME="E1:61_2_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> or <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> )
 ;<br/><a NAME="E2:61_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">j</font> or ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> ) )
 <b>by </b><i><a class="ref" href="waybel_3.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:61_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span> <b>by </b><i><a class="ref" href="waybel_3.html#D2" target="_self">Def2</a>, </i>;<br/><a NAME="E5:61_2"/>
<font color="Maroon">f</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:61_2_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:61_2_2"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">g</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> &amp; <font color="Maroon">g</font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font> )
 
<b>by </b><i>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</a>, <a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><a NAME="E1:61_2_2_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> or <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> )
 ;<br/><a NAME="E2:61_2_2_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">j</font> or ( <font color="Maroon">f</font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> ) )
 <b>by </b><i><a class="ref" href="waybel_3.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:61_2_2_1"/>
<font color="Maroon">f</font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">g</font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T1" target="_self">Th1</a>, , <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:61_2_2"/>
<font color="Maroon">f</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E6:61_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f</font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="waybel_3.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a>, <a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/><b>hence </b><a NAME="E7:61_2"/>
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">i</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:61"/>
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_3.html#D1" target="_self">Def1</a>, , <a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>


</div>
