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<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/>



<a NAME="E1:51"/>
 <a href="waybel24.html#K3" target="_self">ContMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <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>
 
<b>by </b><i/>;<br/>
<a NAME="E2:51"/><b>then </b>
the <a href="struct_0.html#U1">carrier</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> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</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>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">F</font> as   <a href="struct_0.html#NM2">Subset</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>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">sF</font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">F'</font> 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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:51_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><b>set </b><font color="Maroon">G1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">f</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">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F'</font> </span>}</span> ;<br/><a NAME="E2:51_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">sF</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="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">f</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">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F'</font> </span>}</span> ,<font color="Maroon">T</font>
 <b>by </b><i/>;<br/><a NAME="E3:51_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">f</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">f</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">sF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_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="E1:51_1_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="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">f</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">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">f'</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="E3:51_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F'</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f'</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/>
<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f1</font> as  <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/>
<a NAME="E6:51_1_1"/>
<font color="Maroon">f'</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">F'</font>
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<a NAME="E7:51_1_1"/><b>then </b>
<font color="Maroon">f1</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="E8:51_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">sF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T10">YELLOW_2:10</a></i>;<br/>
<a NAME="E9:51_1_1"/>
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:51_1_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">sF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</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><b>hence </b><a NAME="E4:51_1"/>
<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> <font color="Maroon">sF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:51"/>
 <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> is  <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>
 <b>by </b><i><a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a></i>;<br/>


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