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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/>

<a NAME="E1:40"/><i><font color="Green">E36</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K2">SCMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:40_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</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"><a href="waybel17.html#K2">SCMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:40_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font> as  <a href="waybel_0.html#V22">directed-sups-preserving</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="E3:40_1_1"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
;<br/>
<b>hence </b><a NAME="E4:40_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K2">SCMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel17.html#D2">WAYBEL17:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</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="E2:40_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K2">SCMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
 ;<br/>

<a NAME="E3:40_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K2">SCMaps</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"><a href="yellow_1.html#K7">MonMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</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">x</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>, <a class="ref" href="waybel10.html#T10">WAYBEL10:10</a></i>;<br/>
<a NAME="E5:40_1"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="waybel17.html#D2">WAYBEL17:def 2</a></i>;<br/>
<a NAME="E6:40_1"/><b>then </b>
<font color="Maroon">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i><a class="ref" href="waybel17.html#T22">WAYBEL17:22</a></i>;<br/>
<b>hence </b><a NAME="E7:40_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:40"/><b>then </b><i><font color="Green">E39</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</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"><a href="yellow_1.html#K7">MonMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E3:40"/>
 <a href="waybel27.html#K2" target="_self">UPS</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/>
<b>then </b>the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> = 
the <a href="orders_2.html#U1">InternalRel</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> <a href="toler_1.html#K1">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#D14">YELLOW_0:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<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> <a href="toler_1.html#K1">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K7">MonMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="toler_1.html#K1">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
<b>by </b><i>, <a class="ref" href="wellord1.html#T29">WELLORD1:29</a></i>
<br/>.= 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K7">MonMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="toler_1.html#K1">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K2">SCMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_0.html#D14">YELLOW_0:def 14</a></i>
<br/>.= 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K2">SCMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#D14">YELLOW_0:def 14</a></i>
;<br/>
<b>hence </b><a NAME="E5:40"/>
 <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="waybel17.html#K2">SCMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font>
 <b>by </b><i/>;<br/>


</div>
