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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel11.html#V4">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:61"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_8.html#V2">algebraic</a>
 <b>and </b><br/><a NAME="E2:61"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_8.html#V2">algebraic</a>
 ;<br/>

<a NAME="E3:61"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel17.html#E1:61"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_8.html#T7">WAYBEL_8:7</a></i>;<br/>
<a NAME="E4:61"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel17.html#E2:61"><i><font color="Green">E44</font></i></a>, <a class="ref" href="waybel_8.html#T7">WAYBEL_8:7</a></i>;<br/>
<b>assume </b><a NAME="E5:61"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E6:61"/><i><font color="Green">E48</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> is <a href="waybel_3.html#V1">compact</a> );<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> = <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E7:61"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)</span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="waybel17.html#S1" target="_self">WAYBEL17:sch 1</a>(<a class="txt" href="waybel17.html#E6:61"><i><font color="Green">E48</font></i></a>);<br/></i>
<a NAME="E8:61"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_8.html#D2">WAYBEL_8:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>4</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel17.html#E1:61"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_8.html#D4">WAYBEL_8:def 4</a></i>;<br/>
<a NAME="E10:61"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel17.html#E3:61"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel17.html#E4:61"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel17.html#E5:61"><i><font color="Green">E47</font></i></a>, <a class="txt" href="waybel17.html#E42"><i><font color="Green">Lemma37</font></i></a></i>;<br/>
<a NAME="E11:61"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E12:61"/><i><font color="Green">E52</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E13:61"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_8.html#V1">satisfying_axiom_K</a>
 
<b>by </b><i><a class="txt" href="waybel17.html#E1:61"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_8.html#D4">WAYBEL_8:def 4</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_8.html#D3">WAYBEL_8:def 3</a></i>
<br/>.= 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span> ,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="waybel17.html#E7:61"><i><font color="Green">E49</font></i></a>, <a class="txt" href="waybel17.html#E8:61"><i><font color="Green">E50</font></i></a>, <a class="txt" href="waybel17.html#E11:61"><i><font color="Green">E51</font></i></a>, <a class="txt" href="waybel17.html#E12:61"><i><font color="Green">E52</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
;<br/>
<b>hence </b><a NAME="E15:61"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span> ,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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