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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K9" title="PROB_1:func.9">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/>

<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_4.html#T13" target="_self" title="TOPGEN_4:th.13">Th13</a></span>;<br/>
<span class="kw">set </span><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="E2:105"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
 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="105_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:105_1"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E2:105_1"/><span class="lab"><font color="Green" title="E54">A3</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">Z9</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105_1"><span class="lab"><font color="Green" title="E54">A3</font></span></a></span>;<br/>
<a NAME="E4:105_1"/>
<font color="Maroon" title="c5">Z9</font> is  <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="topgen_4.html#T59" target="_self" title="TOPGEN_4:th.59">Th63</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:105_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_4.html#D11" target="_self" title="TOPGEN_4:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:105"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_4.html#T65" target="_self" title="TOPGEN_4:th.65">Th69</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:105"/>
 <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#K9" title="PROB_1:func.9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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