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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V2">TopSpace-like</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="waybel_0.html#V24">up-complete</a> <a href="waybel11.html#V4">Scott</a>  <a href="waybel_9.html#L1">TopRelStr</a> , <font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V2">TopSpace-like</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="waybel_0.html#V24">up-complete</a> <a href="waybel11.html#V4">Scott</a>  <a href="waybel_9.html#L1">TopRelStr</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="E1:13"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 ;<br/>

<a NAME="E2:13"/><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/>;<br/>
<b>thus </b><a NAME="E3:13"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a><br/>

<b>reconsider </b><font color="Maroon">P1'</font> = <font color="Maroon">P1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>assume </b><a NAME="E2:13_1"/>
<font color="Maroon">P1</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E3:13_1"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">P1'</font> is <a href="waybel11.html#V2">closed_under_directed_sups</a> &amp; <font color="Maroon">P1'</font> is <a href="waybel_0.html#V12">lower</a> )
 
<b>by </b><i><a class="ref" href="waybel11.html#T7">WAYBEL11:7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:13_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font>
 ;<br/><a NAME="E2:13_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</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="E3:13_1_1"/>
 <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="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/><a NAME="E4:13_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/><b>reconsider </b><font color="Maroon">fD</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font> as  <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i>, <a class="ref" href="yellow_2.html#T17">YELLOW_2:17</a></i>;<br/><a NAME="E6:13_1_1"/>
<font color="Maroon">fD</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#T5">BORSUK_1:5</a></i>;<br/><a NAME="E7:13_1_1"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">fD</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1'</font>
 <b>by </b><i>, <a class="ref" href="waybel11.html#D2">WAYBEL11:def 2</a></i>;<br/><b>hence </b><a NAME="E8:13_1_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:13_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font> is <a href="waybel11.html#V2">closed_under_directed_sups</a> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font> is <a href="waybel_0.html#V12">lower</a> )
 
<b>by </b><i>, , , <a class="ref" href="waybel11.html#D2">WAYBEL11:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:13_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="waybel11.html#T7">WAYBEL11:7</a></i>;<br/>


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


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