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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>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM8">ManySortedSet</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:2"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> holds  <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> is    <a href="yellow_8.html#M1">Basis</a> of <font color="Olive">x</font>
 ;<br/>

<a NAME="E2:2"/><i><font color="Green">E31</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:2"/>
 <a href="card_3.html#K3">Union</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a>, </i>;<br/>
<a NAME="E5:2_1"/>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">W</font> =  <a href="card_3.html#K3">Union</a> <font color="Maroon">B</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E5:2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</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="2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a>, </i>;<br/>
<a NAME="E5:2_2"/>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:2_2"/><b>then </b>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</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_8.html#D2">YELLOW_8:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:2_2"/>
<font color="Maroon">a</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/>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_3"><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">T</font>;<br/><b>assume </b><a NAME="E1:2_3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E2:2_3"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E3:2_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:2_3"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , <a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>thus </b><a NAME="E6:2_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i>, , <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/><b>thus </b><a NAME="E7:2_3"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/><b>thus </b><a NAME="E8:2_3"/>
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:2"/>
 <a href="card_3.html#K3">Union</a> <font color="Maroon">B</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T32">YELLOW_9:32</a></i>;<br/>


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