<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topgen_4.html#K5" title="TOPGEN_4:func.5">BorelSets</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="openlatt.html#K1" title="OPENLATT:func.1">Topology_of</a> <font color="Maroon" title="c1">T</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c2">BT</font> =  <a href="topgen_4.html#K5" title="TOPGEN_4:func.5">BorelSets</a> <font color="Maroon" title="c1">T</font> as   <a href="topgen_4.html#NM1" title="TOPGEN_4:NM.1">SigmaField</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="topgen_4.html#T13" target="_self" title="TOPGEN_4:th.13">Th13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">X</font> =  <a href="openlatt.html#K1" title="OPENLATT:func.1">Topology_of</a> <font color="Maroon" title="c1">T</font>;<br/>
<a NAME="E2:107"/><i><font color="Green" title="E54">A1</font></i>: 
 <a href="openlatt.html#K1" title="OPENLATT:func.1">Topology_of</a> <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">BT</font>
 
<b>by </b><i><a class="ref" href="topgen_4.html#T69" target="_self" title="TOPGEN_4:th.69">Th69</a></i>;<br/>
<a NAME="E3:107"/>
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st  <a href="openlatt.html#K1" title="OPENLATT:func.1">Topology_of</a> <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font> &amp; <font color="Olive" title="b1">Z</font> is   <a href="topgen_4.html#NM1" title="TOPGEN_4:NM.1">SigmaField</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Maroon" title="c2">BT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">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" title="c4">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="openlatt.html#K1" title="OPENLATT:func.1">Topology_of</a> <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Z</font> &amp; <font color="Maroon" title="c4">Z</font> is   <a href="topgen_4.html#NM1" title="TOPGEN_4:NM.1">SigmaField</a> of <font color="Maroon" title="c1">T</font> implies <font color="Maroon" title="c2">BT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Z</font> )</span><br/>

<b>assume </b><a NAME="E1:107_1"/><i><font color="Green" title="E55">A2</font></i>: 
(  <a href="openlatt.html#K1" title="OPENLATT:func.1">Topology_of</a> <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Z</font> &amp; <font color="Maroon" title="c4">Z</font> is   <a href="topgen_4.html#NM1" title="TOPGEN_4:NM.1">SigmaField</a> of <font color="Maroon" title="c1">T</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">BT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Z</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">Z'</font> = <font color="Maroon" title="c4">Z</font> as   <a href="topgen_4.html#NM1" title="TOPGEN_4:NM.1">SigmaField</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E3:107_1"/>
<font color="Maroon" title="c5">Z'</font> is <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a>
 
<b>by </b><i><a class="txt" href="#E1:107_1"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="topgen_4.html#T63" target="_self" title="TOPGEN_4:th.63">Th63</a></i>;<br/>
<b>hence </b><a NAME="E4:107_1"/>
<font color="Maroon" title="c2">BT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Z</font>
 <b>by </b><i><a class="ref" href="topgen_4.html#D11" target="_self" title="TOPGEN_4:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:107"/>
 <a href="topgen_4.html#K5" title="TOPGEN_4:func.5">BorelSets</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="openlatt.html#K1" title="OPENLATT:func.1">Topology_of</a> <font color="Maroon" title="c1">T</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:107"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="prob_1.html#D14" title="PROB_1:def.14">PROB_1:def 14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
