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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">BT</font> =  <a href="topgen_4.html#K5" target="_self">BorelSets</a> <font color="Maroon">T</font>;<br/>
<b>reconsider </b><font color="Maroon">BT</font> =  <a href="topgen_4.html#K5" target="_self">BorelSets</a> <font color="Maroon">T</font> as   <a href="topgen_4.html#NM2">SigmaField</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">X</font> =  <a href="openlatt.html#K1">Topology_of</a> <font color="Maroon">T</font>;<br/>
<a NAME="E2:107"/><i><font color="Green">E37</font></i>: 
 <a href="openlatt.html#K1">Topology_of</a> <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">BT</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:107"/>
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st  <a href="openlatt.html#K1">Topology_of</a> <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font> &amp; <font color="Olive">Z</font> is   <a href="topgen_4.html#NM2">SigmaField</a> of <font color="Maroon">T</font> holds <br/><font color="Maroon">BT</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:107_1"/><i><font color="Green">E39</font></i>: 
(  <a href="openlatt.html#K1">Topology_of</a> <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> is   <a href="topgen_4.html#NM2">SigmaField</a> of <font color="Maroon">T</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">Z'</font> = <font color="Maroon">Z</font> as   <a href="topgen_4.html#NM2">SigmaField</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E3:107_1"/>
<font color="Maroon">Z'</font> is <a href="topgen_4.html#V1" target="_self">all-open-containing</a>
 
<b>by </b><i><a class="ref" href="topgen_4.html#T5" target="_self">Th5</a>, </i>;<br/>
<b>hence </b><a NAME="E4:107_1"/>
<font color="Maroon">BT</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:107"/>
 <a href="topgen_4.html#K5" target="_self">BorelSets</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="prob_1.html#K11">sigma</a> <span class="p1">(<span class="default"><a href="openlatt.html#K1">Topology_of</a> <font color="Maroon">T</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="prob_1.html#D14">PROB_1:def 14</a></i>;<br/>


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