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<div class="add">

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V24">up-complete</a> <a href="waybel11.html#V4">Scott</a> <a href="waybel19.html#NM1">TopPoset</a>;<br/><b>let </b><font color="Maroon">F</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 <span class="p1">(<span class="default"><a href="waybel24.html#K3">ContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span>;<br/>





<b>set </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">L</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span>;<br/>
<a NAME="E1:8"/>
 <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T28">YELLOW_1:28</a></i>;<br/>
<b>then 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">L</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">L</font> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E3:8"/>
 <a href="waybel24.html#K3">ContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">L</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="waybel24.html#D3">WAYBEL24:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">aF</font> = <font color="Maroon">F</font> as  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 <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_2.html#T7">YELLOW_2:7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E2:8_1"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">A</font> is <a href="waybel_0.html#V13">upper</a> &amp; <font color="Maroon">A</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> )
 <b>by </b><i><a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:8_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sF</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:8_1_1_1"/><b>then </b><i><font color="Green">E50</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">sF</font> &amp; <font color="Maroon">sF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E3:8_1_1_1"/><i><font color="Green">E53</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">sF</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E5:8_1_1_1"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="waybel_5.html#K7">=&gt;</a> <font color="Maroon">L</font></span>)</span> <a href="yellow16.html#K2">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E6:8_1_1_1"/><i><font color="Green">E72</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">aF</font>,<font color="Maroon">L</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="waybel_5.html#K7">=&gt;</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a>, <a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/><a NAME="E7:8_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">aF</font></span>)</span> <a href="waybel27.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="yellow16.html#K1">pi</a> <font color="Maroon">aF</font>,<font color="Maroon">y</font></span>)</span> &amp;  <a href="yellow16.html#K1">pi</a> <font color="Maroon">aF</font>,<font color="Maroon">y</font> is <a href="waybel_0.html#V1">directed</a> &amp; not  <a href="yellow16.html#K1">pi</a> <font color="Maroon">aF</font>,<font color="Maroon">y</font> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i>, <a class="ref" href="yellow16.html#T35">YELLOW16:35</a>, <a class="ref" href="yellow16.html#T37">YELLOW16:37</a></i>;<br/><a NAME="E8:8_1_1_1"/><b>then </b>
 <a href="yellow16.html#K1">pi</a> <font color="Maroon">aF</font>,<font color="Maroon">y</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="waybel11.html#D1">WAYBEL11:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:8_1_1_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="yellow16.html#K1">pi</a> <font color="Maroon">aF</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E12:8_1_1_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">aF</font> &amp; <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">y</font> )
 <b>by </b><i>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/><b>reconsider </b><font color="Maroon">f</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">X</font>,<font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="waybel24.html#T21">WAYBEL24:21</a></i>;<br/><a NAME="E14:8_1_1_1"/><i><font color="Green">E76</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>take </b><font color="Maroon">Q</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font>;<br/><b>thus </b><a NAME="E15:8_1_1_1"/>
<font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/><b>thus </b><a NAME="E16:8_1_1_1"/>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">sF</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:8_1_1_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/>

<a NAME="E2:8_1_1_1_1"/><b>then </b><i><font color="Green">E78</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E4:8_1_1_1_1"/>
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="waybel_5.html#K7">=&gt;</a> <font color="Maroon">L</font></span>)</span> <a href="yellow16.html#K2">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E5:8_1_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
( <font color="Maroon">sF</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="yellow16.html#K1">pi</a> <font color="Maroon">aF</font>,<font color="Maroon">b</font></span>)</span> &amp;  <a href="yellow16.html#K1">pi</a> <font color="Maroon">aF</font>,<font color="Maroon">b</font> is <a href="waybel_0.html#V1">directed</a> &amp; not  <a href="yellow16.html#K1">pi</a> <font color="Maroon">aF</font>,<font color="Maroon">b</font> is <a href="xboole_0.html#V1">empty</a> )
 
<b>by </b><i>, <a class="ref" href="yellow16.html#T35">YELLOW16:35</a>, <a class="ref" href="yellow16.html#T37">YELLOW16:37</a></i>;<br/>
<a NAME="E6:8_1_1_1_1"/><b>then </b>
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="yellow16.html#K1">pi</a> <font color="Maroon">aF</font>,<font color="Maroon">b</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E7:8_1_1_1_1"/><b>then </b>
( <font color="Maroon">sF</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="yellow16.html#K1">pi</a> <font color="Maroon">aF</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="yellow16.html#K1">pi</a> <font color="Maroon">aF</font>,<font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="ref" href="card_3.html#D6">CARD_3:def 6</a>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>
<a NAME="E8:8_1_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">sF</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E9:8_1_1_1_1"/><b>then </b>
<font color="Maroon">sF</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>
<b>hence </b><a NAME="E10:8_1_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sF</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E17:8_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div>
<b>end;</b></div><b>thus </b><a NAME="E2:8_1_1"/>
(  ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Q</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">sF</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Q</font> ) implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sF</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:8_1"/>
<font color="Maroon">sF</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T57">TOPS_1:57</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:8"/>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">F</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


</div>
