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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>set </b><font color="Maroon">A</font> = the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>;<br/>
<b>reconsider </b><font color="Maroon">IA</font> =  <a href="yellow_1.html#K2" target="_self">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> as  <a href="yellow_0.html#V1">lower-bounded</a> <a href="orders_2.html#NM1">Poset</a> <b>by </b><i/>;<br/>
<a NAME="E2:37"/>
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">IA</font> holds   <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">X</font>, <a href="yellow_1.html#K2" target="_self">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">IA</font>;<br/>

<b>set </b><font color="Maroon">N</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>;<br/>
<b>reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E2:37_1"/>
<font color="Maroon">X'</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
;<br/>
<a NAME="E3:37_1"/><b>then </b>
 <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">N</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> ;<br/>
<a NAME="E5:37_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span>; <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="E1:37_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:37_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1_1"/>
<font color="Maroon">N</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:37_1"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span>  st <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Maroon">N</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:37_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">b</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_1_2_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon">Z1'</font> = <font color="Maroon">Z1</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> ;<br/><a NAME="E3:37_1_2_1"/>
<font color="Maroon">Z1'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><b>hence </b><a NAME="E4:37_1_2_1"/>
<font color="Maroon">Z1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E3:37_1_2"/><b>then </b>
 <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/>
<b>hence </b><a NAME="E4:37_1_2"/>
<font color="Maroon">N</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:37_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>, <a href="yellow_1.html#K2" target="_self">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T15">YELLOW_0:15</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:37"/>
 <a href="yellow_1.html#K2" target="_self">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> is <a href="lattice3.html#V3">complete</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#T53">YELLOW_0:53</a></i>;<br/>


</div>
