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<b>let </b><font color="Maroon" title="c2">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies <font color="Maroon" title="c2">S</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> )</span><br/>

<b>assume </b><a NAME="E1:5_1_1"/><i><font color="Green" title="E3">Z</font></i>: 
<font color="Maroon" title="c2">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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">S</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a></span><br/>

<b>let </b><font color="Maroon" title="c3">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D7" target="_self" title="COMPTS_1:def.7">COMPTS_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c3">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> implies  ex <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Olive" title="b1">G</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b1">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:5_1_1"/>
<font color="Maroon" title="c3">F</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c2">S</font>
 <b>and </b><br/><a NAME="E3:5_1_1"/>
<font color="Maroon" title="c3">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Olive" title="b1">G</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b1">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c4">C</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c4">C</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c4">C</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/>

<a NAME="E5:5_1_1"/>
<font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a>, <a class="txt" href="#E1:5_1_1"><i><font color="Green" title="E3">Z</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:5_1_1"/>
( <font color="Maroon" title="c4">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c4">C</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c4">C</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>; <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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