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<b>reconsider </b><font color="Maroon" title="c3">w</font> = <font color="Maroon" title="c2">w</font> as  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>  <b>by </b><i><a class="txt" href="waybel34.html#E1:25"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">r</font> being  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> <a href="yellow21.html#V5" title="YELLOW21:attr.5">upper-bounded</a> <a href="orders_1.html#NM2" title="ORDERS_1:NM.2">Order</a> of <font color="Maroon" title="c3">w</font>;<br/>
<a NAME="E3:25_1_1"/>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">r</font> #) is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>
 
;<br/>
<a NAME="E4:25_1_1"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
 ex <font color="Olive" title="b1">x</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> )
 
;<br/>
<a NAME="E5:25_1_1"/><i><font color="Green" title="E17">A3</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">b</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">c</font>] holds <br/>for <font color="Olive" title="b4">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/> for <font color="Olive" title="b5">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">f</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b5">g</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">f</font>]
 
<b>by </b><i><a class="ref" href="waybel20.html#T28" title="WAYBEL20:th.28">WAYBEL20:28</a></i>;<br/>
<a NAME="E6:25_1_1"/><i><font color="Green" title="E18">A4</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>, <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Olive" title="b1">a</font>]
 
;<br/>
<b>thus </b><a NAME="E7:25_1_1"/>
 ex <font color="Olive" title="b1">C</font> being <a href="altcat_1.html#V6" title="ALTCAT_1:attr.6">strict</a> <a href="yellow21.html#V2" title="YELLOW21:attr.2">lattice-wise</a> <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a> st <br/>( ( for <font color="Olive" title="b2">x</font> being  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> holds <br/> ( <font color="Olive" title="b2">x</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Olive" title="b1">C</font> iff ( <font color="Olive" title="b2">x</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> ) ) ) &amp; ( for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Olive" title="b1">C</font><br/> for <font color="Olive" title="b4">f</font> being <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Olive" title="b2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Olive" title="b3">b</font></span>)</span> holds <br/> ( <font color="Olive" title="b4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <font color="Maroon">S<sub>2</sub></font>[ <a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Olive" title="b2">a</font>, <a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">f</font>] ) ) )
 <b>from </b><i><a class="ref" href="yellow21.html#S3" title="YELLOW21:sch.3">YELLOW21:sch 3</a>(<a class="txt" href="waybel34.html#E4:25_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="waybel34.html#E5:25_1_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="waybel34.html#E6:25_1_1"><i><font color="Green" title="E18">A4</font></i></a>);</i><br/>


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