<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">S</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel11.html#V4">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a>, <font color="Maroon">T</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">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="E1:63"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">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="Olive">x</font> &amp; <font color="Olive">w</font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span> ,<font color="Maroon">T</font>
 ;<br/>

<a NAME="E2:63"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">f</font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i><a class="txt" href="waybel17.html#E1:63"><i><font color="Green">E43</font></i></a></i>;<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/>

<a NAME="E3:63"/><i><font color="Green">E47</font></i>: 
<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> &amp; <font color="Olive">w</font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span> ,<font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">FW</font> = <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> &amp; <font color="Olive">w</font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">FX</font> = <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> ;<br/>
<b>set </b><font color="Maroon">fx</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>;<br/>
<a NAME="E4:63"/><i><font color="Green">E48</font></i>: 
<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> &amp; <font color="Olive">w</font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span>  <a href="tarski.html#R1">c=</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:63_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</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> &amp; <font color="Olive">w</font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">w</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:63_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">w</font> is <a href="waybel_3.html#V1">compact</a> )
 ;<br/>
<a NAME="E4:63_1"/>
<font color="Maroon">w</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">w</font>
 
<b>by </b><i><a class="txt" href="waybel17.html#E3:63_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="waybel_3.html#D2">WAYBEL_3:def 2</a></i>;<br/>
<a NAME="E5:63_1"/><b>then </b>
<font color="Maroon">w</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="waybel17.html#E3:63_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="waybel_3.html#T2">WAYBEL_3:2</a></i>;<br/>
<b>hence </b><a NAME="E6:63_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</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> 
 <b>by </b><i><a class="txt" href="waybel17.html#E3:63_1"><i><font color="Green">E49</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:63"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:63_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</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> 
 ;<br/>

<b>then consider </b><font color="Maroon">ww</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:63_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ww</font> &amp; <font color="Maroon">ww</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font> )
 ;<br/>
<a NAME="E4:63_2"/>
<font color="Maroon">ww</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="waybel_3.html#T1">WAYBEL_3:1</a></i>;<br/>
<b>hence </b><a NAME="E5:63_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="waybel17.html#E2:63"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:63"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  st <font color="Olive">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</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>  holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:63_3"/>
<font color="Maroon">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</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> 
 ;<br/>

<a NAME="E2:63_3"/><b>then </b>
<font color="Maroon">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</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> &amp; <font color="Olive">w</font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel17.html#E4:63"><i><font color="Green">E48</font></i></a>, <a class="ref" href="yellow_0.html#T9">YELLOW_0:9</a></i>;<br/>
<b>hence </b><a NAME="E3:63_3"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="waybel17.html#E3:63"><i><font color="Green">E47</font></i></a>, <a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:63"/>
<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>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>


</div>
