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<b>let </b><font color="Maroon">S</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="lattice3.html#V3">complete</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="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<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/>





<i><font color="Green">E27</font></i>:  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span></span>)</span> = 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<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_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_0.html#T34">WAYBEL_0:34</a></i>
;<br/>
<a NAME="E2:9"/><i><font color="Green">E43</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/>
<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/>
<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="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>;<br/>
<a NAME="E3:9"/><i><font color="Green">E44</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="waybel24.html#S1" target="_self">WAYBEL24:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>] <b>means </b> ex <font color="Olive">y1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y1</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> );<br/>
<a NAME="E4:9"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x2</font>] iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x2</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:9_2_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">x2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:9_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:9_2_1"/>
 ex <font color="Olive">y1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Maroon">x2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y1</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">y1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><a NAME="E2:9_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>

<b>thus </b><a NAME="E3:9_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x2</font>]
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:9"/><i><font color="Green">E60</font></i>: 
<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"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x1</font>) where <font color="Olive">x1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x1</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="waybel24.html#S2" target="_self">WAYBEL24:sch 2</a>();<br/></i>
<a NAME="E6:9"/>
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <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="Olive">w</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T14">WAYBEL_0:14</a></i>;<br/>
<b>hence </b><a NAME="E7:9"/>
<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="orders_2.html#R3">&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#T34">WAYBEL_0:34</a></i>;<br/>


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