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<b>let </b><font color="Maroon">S</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">T</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:34"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>) <b>-&gt; </b>  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> = <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:34"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">f</font> <a href="waybel_0.html#R4">preserves_sup_of</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E3:34"/><i><font color="Green">E47</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font>,<font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E4:34"/><i><font color="Green">E48</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:34"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</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">w</font>) where <font color="Olive">w</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">w</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">w</font>)</span>)</span> where <font color="Olive">w</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">w</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="waybel17.html#S2" target="_self">WAYBEL17:sch 2</a>();<br/></i>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">w</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font> </span>}</span> ,<font color="Maroon">T</font>
<b>by </b><i>, , <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a>, </i>
;<br/>
<b>hence </b><a NAME="E7:34"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">w</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font> </span>}</span> ,<font color="Maroon">T</font>
 ;<br/>


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