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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ( <font color="Maroon">a<sub>1</sub></font> is   <a href="waybel_0.html#NM8">LATTICE</a> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] );<br/>
<b>consider </b><font color="Maroon">c<sub>1</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:20_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow21.html#K2" target="_self">POSETS</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being  <a href="orders_2.html#V1">strict</a> <a href="waybel_0.html#NM8">LATTICE</a><b> such that </b><br/><a NAME="E4:20_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 <b>by </b><i><a class="txt" href="yellow21.html#E1:20"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E5:20_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="yellow21.html#K1" target="_self">as_1-sorted</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="yellow21.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E6:20_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow21.html#K2" target="_self">POSETS</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="yellow21.html#E4:20_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="yellow21.html#D2" target="_self">Def2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="yellow21.html#E2:20_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yellow21.html#E4:20_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E8:20_1"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <font color="Olive">b<sub>1</sub></font> is   <a href="waybel_0.html#NM8">LATTICE</a>
 
<b>by </b><i><a class="txt" href="yellow21.html#E2:20_1"><i><font color="Green">E12</font></i></a></i>;<br/>
<div><i><font color="Green">E15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>assume </b><a NAME="E1:20_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><a NAME="E2:20_1_1"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>] )
 <b>by </b><i><a class="txt" href="yellow21.html#E2:20_1"><i><font color="Green">E12</font></i></a></i>;<br/><b>hence </b><a NAME="E3:20_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> st <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="yellow21.html#E2:20"><i><font color="Green">E10</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>assume </b><a NAME="E1:20_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:20_1_2"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="yellow21.html#E2:20_1"><i><font color="Green">E12</font></i></a></i>;<br/><b>hence </b><a NAME="E3:20_1_2"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="yellow21.html#E3:20"><i><font color="Green">E11</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being  <a href="altcat_1.html#V6">strict</a> <a href="yellow21.html#V2" target="_self">lattice-wise</a> <a href="altcat_1.html#NM4">category</a><b> such that </b><br/><a NAME="E12:20_1"/><i><font color="Green">E17</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E13:20_1"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>4</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">b<sub>2</sub></font></span>)</span> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Maroon">P<sub>2</sub></font>[ <a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">b<sub>1</sub></font>, <a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] )
 <b>from </b><i><a class="ref" href="yellow21.html#S2" target="_self">YELLOW21:sch 2</a>(<a class="txt" href="yellow21.html#E8:20_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow21.html#E9:20_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="yellow21.html#E10:20_1"><i><font color="Green">E16</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/><a NAME="E1:20_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="yellow21.html#K1" target="_self">as_1-sorted</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow21.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E2:20_1_3"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow21.html#K2" target="_self">POSETS</a> <font color="Maroon">F<sub>1</sub></font>() iff ( <font color="Maroon">c<sub>5</sub></font> is  <a href="orders_2.html#V1">strict</a> <a href="orders_2.html#NM1">Poset</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() ) )
 <b>by </b><i><a class="ref" href="yellow21.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E3:20_1_3"/>
( <font color="Maroon">c<sub>5</sub></font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>4</sub></font> iff ( <font color="Maroon">c<sub>5</sub></font> is <a href="orders_2.html#V1">strict</a> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>] &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() ) )
 <b>by </b><i><a class="txt" href="yellow21.html#E2:20_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yellow21.html#E12:20_1"><i><font color="Green">E17</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E15:20_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>4</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">b<sub>2</sub></font></span>)</span> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Maroon">P<sub>2</sub></font>[ <a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">b<sub>1</sub></font>, <a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] )
 <b>by </b><i><a class="txt" href="yellow21.html#E13:20_1"><i><font color="Green">E18</font></i></a></i>;<br/>


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