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<b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">w</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">r</font> being  <a href="wellord1.html#V2">well-ordering</a> <a href="yellow21.html#V5" target="_self">upper-bounded</a> <a href="orders_1.html#NM2">Order</a> of <font color="Maroon">w</font>;<br/>
<a NAME="E3:48_1_1"/>
 <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">w</font>,<font color="Maroon">r</font> #) is <a href="lattice3.html#V3">complete</a>
 
;<br/>
<a NAME="E4:48_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
 ex <font color="Olive">x</font> being <a href="orders_2.html#V1">strict</a> <a href="waybel_0.html#NM8">LATTICE</a> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 
;<br/>
<a NAME="E5:48_1_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="waybel_0.html#NM8">LATTICE</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] holds <br/>for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Olive">a</font>,<font color="Olive">b</font><br/> for <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Olive">b</font>,<font color="Olive">c</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b</font>,<font color="Olive">c</font>,<font color="Olive">g</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>,<font color="Olive">c</font>,<font color="Olive">g</font> <a href="partfun1.html#K1">*</a> <font color="Olive">f</font>]
 
<b>by </b><i><a class="ref" href="waybel20.html#T29">WAYBEL20:29</a></i>;<br/>
<a NAME="E6:48_1_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">a</font> being  <a href="waybel_0.html#NM8">LATTICE</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>,<font color="Olive">a</font>, <a href="grcat_1.html#K7">id</a> <font color="Olive">a</font>]
 
;<br/>
<b>thus </b><a NAME="E7:48_1_1"/>
 ex <font color="Olive">It</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> st <br/>( ( for <font color="Olive">x</font> being  <a href="waybel_0.html#NM8">LATTICE</a> holds <br/> ( <font color="Olive">x</font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Olive">It</font> iff ( <font color="Olive">x</font> is <a href="orders_2.html#V1">strict</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> ) ) ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Olive">It</font><br/> for <font color="Olive">f</font> being <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">b</font></span>)</span> holds <br/> ( <font color="Olive">f</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">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Maroon">S<sub>2</sub></font>[ <a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">a</font>, <a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">b</font>,<font color="Olive">f</font>] ) ) )
 <b>from </b><i><a class="ref" href="yellow21.html#S3" target="_self">YELLOW21:sch 3</a>(, , );</i><br/>


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