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<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:61"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c1">S</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 <b>and </b><br/><a NAME="E2:61"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c2">T</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 ;<br/>

<a NAME="E3:61"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c1">S</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel17.html#E1:61"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="waybel_8.html#T7" title="WAYBEL_8:th.7">WAYBEL_8:7</a></i>;<br/>
<a NAME="E4:61"/><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon" title="c2">T</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel17.html#E2:61"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="waybel_8.html#T7" title="WAYBEL_8:th.7">WAYBEL_8:7</a></i>;<br/>
<b>assume </b><a NAME="E5:61"/><i><font color="Green" title="E47">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">w</font></span>)</span> where <font color="Olive" title="b2">w</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> : <font color="Olive" title="b2">w</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b1">x</font> </span>}</span> ,<font color="Maroon" title="c2">T</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>

<a NAME="E6:61"/><i><font color="Green" title="E48">A6</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>] <b>means </b>( $1 <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font> &amp; $1 is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> );<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>) <b>-&gt; </b>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> = $1;<br/>
<a NAME="E7:61"/><i><font color="Green" title="E49">A7</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">w</font>) where <font color="Olive" title="b1">w</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">w</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">w</font>)</span>)</span> where <font color="Olive" title="b1">w</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">w</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="waybel17.html#S1" target="_self" title="WAYBEL17:sch.1">WAYBEL17:sch 1</a>(<a class="txt" href="waybel17.html#E6:61"><i><font color="Green" title="E48">A6</font></i></a>);<br/></i>
<a NAME="E8:61"/><i><font color="Green" title="E50">A8</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> : ( <font color="Olive" title="b1">w</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b1">w</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_8.html#D2" title="WAYBEL_8:def.2">WAYBEL_8:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">D</font> =  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">x</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="waybel17.html#E1:61"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="waybel_8.html#D4" title="WAYBEL_8:def.4">WAYBEL_8:def 4</a></i>;<br/>
<a NAME="E10:61"/>
<font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel17.html#E3:61"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="waybel17.html#E4:61"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="waybel17.html#E5:61"><i><font color="Green" title="E47">A5</font></i></a>, <a class="txt" href="waybel17.html#E42"><i><font color="Green" title="E37">Lm16</font></i></a></i>;<br/>
<a NAME="E11:61"/><b>then </b><i><font color="Green" title="E51">A9</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c5">D</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E12:61"/><i><font color="Green" title="E52">A10</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>
<a NAME="E13:61"/>
<font color="Maroon" title="c1">S</font> is <a href="waybel_8.html#V1" title="WAYBEL_8:attr.1">satisfying_axiom_K</a>
 
<b>by </b><i><a class="txt" href="waybel17.html#E1:61"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="waybel_8.html#D4" title="WAYBEL_8:def.4">WAYBEL_8:def 4</a></i>;<br/>
<b>then </b><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> = 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_8.html#D3" title="WAYBEL_8:def.3">WAYBEL_8:def 3</a></i>
<br/>.= 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> : ( <font color="Olive" title="b1">w</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b1">w</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> ) </span>}</span> ,<font color="Maroon" title="c2">T</font>
<b>by </b><i><a class="txt" href="waybel17.html#E7:61"><i><font color="Green" title="E49">A7</font></i></a>, <a class="txt" href="waybel17.html#E8:61"><i><font color="Green" title="E50">A8</font></i></a>, <a class="txt" href="waybel17.html#E11:61"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="waybel17.html#E12:61"><i><font color="Green" title="E52">A10</font></i></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></i>
;<br/>
<b>hence </b><a NAME="E15:61"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> : ( <font color="Olive" title="b1">w</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b1">w</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> ) </span>}</span> ,<font color="Maroon" title="c2">T</font>
 ;<br/>


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