<?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  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3" target="_self">ContMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>;<br/><b>let </b><font color="Maroon">D</font> be  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">S</font>;<br/>





<b>set </b><font color="Maroon">F'</font> = <font color="Maroon">F</font>;<br/>
<b>set </b><font color="Maroon">L</font> =  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> ,<font color="Maroon">T</font></span>)</span> where <font color="Olive">g</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ,<font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">P</font> =  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">g'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i'</font></span>)</span> where <font color="Olive">g'</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">g'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ,<font color="Maroon">T</font></span>)</span> where <font color="Olive">i'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">i'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> ,<font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">L1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> ,<font color="Maroon">T</font></span>)</span> where <font color="Olive">g</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">P1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i3</font></span>)</span> where <font color="Olive">g2</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ,<font color="Maroon">T</font></span>)</span> where <font color="Olive">i3</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">i3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon">L</font> =  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> ,<font color="Maroon">T</font></span>)</span> where <font color="Olive">g</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ,<font color="Maroon">T</font>, <font color="Maroon">P</font> =  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">g'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i'</font></span>)</span> where <font color="Olive">g'</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">g'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ,<font color="Maroon">T</font></span>)</span> where <font color="Olive">i'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">i'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> ,<font color="Maroon">T</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">sF</font> =  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">F</font>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E3:52"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">sF</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><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="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>;<br/>
<a NAME="E4:52"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> ,<font color="Maroon">T</font></span>)</span> where <font color="Olive">g</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><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="E1:52_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> ,<font color="Maroon">T</font></span>)</span> where <font color="Olive">g</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">g'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span><b> such that </b><br/><a NAME="E3:52_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">b</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">g'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> ,<font color="Maroon">T</font> &amp; <font color="Maroon">g'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g'</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> <b>by </b><i>, </i>;<br/>
<a NAME="E5:52_1"/>
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">g1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:52_1"/><b>then </b><i><font color="Green">E44</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="relat_1.html#K1">dom</a> <font color="Maroon">g'</font>
 
;<br/>
<a NAME="E7:52_1"/>
<font color="Maroon">g'</font> <a href="relat_1.html#K9">.:</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i2</font>) where <font color="Olive">i2</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">i2</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)</span>)</span> where <font color="Olive">i</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">i</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="waybel24.html#S5" target="_self">WAYBEL24:sch 5</a>();<br/></i>
<a NAME="E8:52_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">T</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:52_1"/>
<font color="Maroon">g'</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">F</font>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<a NAME="E10:52_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">g1</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">sF</font>
 
<b>by </b><i><a class="ref" href="waybel10.html#T12">WAYBEL10:12</a></i>;<br/>
<a NAME="E11:52_1"/>
<font color="Maroon">g1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">sF</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:52_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_1_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:52_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E6:52_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">sF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font>
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#T10">YELLOW_2:10</a></i>;<br/>
<a NAME="E7:52_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">sF</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:52_1_1"/><b>then </b>
<font color="Maroon">sF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sF</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E9:52_1_1"/><b>then </b>
<font color="Maroon">sF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">sF</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<b>hence </b><a NAME="E10:52_1_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">sF</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:52_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="waybel24.html#E3:52"><i><font color="Green">E27</font></i></a>, <a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:52"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> =  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">i4</font></span>)</span> where <font color="Olive">i4</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">i4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> ,<font color="Maroon">T</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>;<br/>
<a NAME="E6:52"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">g8</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g8</font>] holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive">g8</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">g8</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:52_2"/>
<font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g1</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E3:52_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">g'</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E4:52_2"/><i><font color="Green">E88</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D</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="E5:52_2"/>
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">g'</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:52_2"/><b>then </b><i><font color="Green">E89</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="relat_1.html#K1">dom</a> <font color="Maroon">g1</font>
 
;<br/>
<a NAME="E7:52_2"/>
<font color="Maroon">g1</font> <a href="relat_1.html#K9">.:</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i2</font>) where <font color="Olive">i2</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">i2</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)</span>)</span> where <font color="Olive">i</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">i</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="waybel24.html#S5" target="_self">WAYBEL24:sch 5</a>();<br/></i>
<b>then </b> <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> ,<font color="Maroon">T</font> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel24.html#E3:52"><i><font color="Green">E27</font></i></a>, , <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
;<br/>
<b>hence </b><a NAME="E9:52_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">g1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">g1</font>)
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:52"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">g3</font>) where <font color="Olive">g3</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g3</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">g4</font>) where <font color="Olive">g4</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g4</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="waybel24.html#S4" target="_self">WAYBEL24:sch 4</a>();<br/></i>
<a NAME="E8:52"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sF</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">D</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:52"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i3</font></span>)</span> where <font color="Olive">g2</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ,<font color="Maroon">T</font></span>)</span> where <font color="Olive">i3</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">i3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span>  <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_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>; <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="E1:52_3"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i3</font></span>)</span> where <font color="Olive">g2</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ,<font color="Maroon">T</font></span>)</span> where <font color="Olive">i3</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">i3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:52_3"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">b</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="Olive">g'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span> where <font color="Olive">g'</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">g'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ,<font color="Maroon">T</font> &amp; <font color="Maroon">i'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 ;<br/>
<a NAME="E4:52_3"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i'</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:52_3"/>
<font color="Maroon">sF</font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:52_3"/><b>then </b><i><font color="Green">E96</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">sF</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="waybel24.html#E3:52"><i><font color="Green">E27</font></i></a>, <a class="ref" href="waybel17.html#T15">WAYBEL17:15</a></i>;<br/>
<a NAME="E7:52_3"/>
<font color="Maroon">i'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
;<br/>
<a NAME="E8:52_3"/><b>then </b>
<font color="Maroon">i'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">sF</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:52_3"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sF</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E10:52_3"/><b>then </b>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">sF</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<b>hence </b><a NAME="E11:52_3"/>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:52"/><b>then </b>
<font color="Maroon">P</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<b>hence </b><a NAME="E11:52"/>
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> ,<font color="Maroon">T</font></span>)</span> where <font color="Olive">g</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ,<font color="Maroon">T</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"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">g'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i'</font></span>)</span> where <font color="Olive">g'</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">g'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ,<font color="Maroon">T</font></span>)</span> where <font color="Olive">i'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">i'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> ,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/>


</div>
