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<b>let </b><font color="Maroon" title="c1">T</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"> for <font color="Olive" title="b1">F</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( <font color="Olive" title="b1">F</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">T</font> iff ( <font color="Olive" title="b1">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> &amp; <font color="Olive" title="b1">F</font> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> ) )</span><br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</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">F</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">T</font> iff ( <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> ) )</span><br/>



<b>thus </b><a NAME="E1:27"/>
( <font color="Maroon" title="c2">F</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">T</font> implies ( <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> implies <font color="Maroon" title="c2">F</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">T</font> )</span><br/>

<b>assume </b><a NAME="E2:27"/><i><font color="Green" title="E13">A1</font></i>: 
( <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">F</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">T</font></span><br/>

<a NAME="E3:27"/>
<font color="Maroon" title="c2">F</font> is <a href="measure1.html#V2" title="MEASURE1:attr.2">sigma-additive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="measure1.html#D9" title="MEASURE1:def.9">MEASURE1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> or  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> )</span><br/>

<b>assume </b><a NAME="E1:27_1"/>
<font color="Maroon" title="c3">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/>

<b>hence </b><a NAME="E2:27_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:27"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="topgen_4.html#D4" target="_self" title="TOPGEN_4:def.4">Def4</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>then reconsider </b><font color="Maroon" title="c3">F</font> = <font color="Maroon" title="c2">F</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="measure1.html#V2" title="MEASURE1:attr.2">sigma-additive</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="topgen_4.html#E2:27"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="topgen_4.html#T12" target="_self" title="TOPGEN_4:th.12">Th12</a></i>;<br/>
<a NAME="E5:27"/>
<font color="Maroon" title="c3">F</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">T</font>
 
;<br/>
<b>hence </b><a NAME="E6:27"/>
<font color="Maroon" title="c2">F</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">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"> verum</span><br/>


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