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<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>;<br/>

<b>set </b><font color="Maroon" title="c3">L</font> =  <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c1">S</font> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">T</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span>)</span> <b>by </b><i><a class="ref" href="waybel24.html#T21" target="_self" title="WAYBEL24:th.21">Th21</a></i>;<br/>
<a NAME="E2:60"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="yellow_0.html#T6" title="YELLOW_0:th.6">YELLOW_0:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">f'</font> = <font color="Maroon" title="c4">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> ;<br/>
<a NAME="E4:60"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span>)</span>  st <font color="Olive" title="b1">b</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><font color="Maroon" title="c4">f</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:60_1"/>
<font color="Maroon" title="c6">b</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">b'</font> = <font color="Maroon" title="c6">b</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> <b>by </b><i><a class="ref" href="waybel24.html#T21" target="_self" title="WAYBEL24:th.21">Th21</a></i>;<br/>
<a NAME="E3:60_1"/>
for <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>

<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> = 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="ref" href="borsuk_1.html#D2" title="BORSUK_1:def.2">BORSUK_1:def 2</a></i>
<br/>.= 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">T</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E2:60_1_1"/><b>then </b>
<font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">b'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">i</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>;<br/>
<b>hence </b><a NAME="E3:60_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:60_1"/>
<font color="Maroon" title="c4">f</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="ref" href="waybel24.html#T20" target="_self" title="WAYBEL24:th.20">Th20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:60"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel24.html#E2:60"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></i>;<br/>
<b>hence </b><a NAME="E6:60"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#D11" title="YELLOW_0:def.11">YELLOW_0:def 11</a></i>;<br/>


</div>
