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

<span class="kw">let </span><font color="Maroon" title="c1">T</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Maroon" title="c1">T</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  &amp; <font color="Olive" title="b1">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  holds <br/><font color="Olive" title="b1">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  &amp; <font color="Maroon" title="c2">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:13"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c1">T</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> 
 <span class="kw">and </span><br/><a NAME="E2:13"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon" title="c2">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 ; <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">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">pp</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="compts_1.html#D4" target="_self" title="COMPTS_1:def.4">COMPTS_1:def 4</a> <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="c4">F</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c4">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="c4">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">P</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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:13"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c4">F</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c2">P</font>
 <span class="kw">and </span><br/><a NAME="E5:13"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c4">F</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c4">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">P</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/>

<span class="kw">set </span><font color="Maroon" title="c5">FP</font> = <font color="Maroon" title="c4">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">pp</font>;<br/>
<a NAME="E6:13"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="tops_1.html#T3" title="TOPS_1:th.3">TOPS_1:3</a></span>;<br/>
<a NAME="E7:13"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
<font color="Maroon" title="c4">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">pp</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">H</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">pp</font> or <font color="Maroon" title="c6">H</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:13_1"/>
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">pp</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="c6">H</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<a NAME="E2:13_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> or <font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:13_1"/>
<font color="Maroon" title="c6">H</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E6:13"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</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>
<span class="kw">reconsider </span><font color="Maroon" title="c6">M</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E9:13"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T40" title="XBOOLE_1:th.40">XBOOLE_1:40</a></span>;<br/>
<a NAME="E10:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E11:13"/>
<font color="Maroon" title="c2">P</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">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="setfam_1.html#D11" title="SETFAM_1:def.11">SETFAM_1:def 11</a></span>;<br/>
<a NAME="E12:13"/><span class="kw">then </span>
<font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></span>;<br/>
<a NAME="E13:13"/><span class="kw">then </span>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T2" title="PRE_TOPC:th.2">PRE_TOPC:2</a></span>;<br/>
<span class="kw">then </span> <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T25" title="ZFMISC_1:th.25">ZFMISC_1:25</a></span>
<br/>.= 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T78" title="ZFMISC_1:th.78">ZFMISC_1:78</a></span>
;<br/>
<a NAME="E15:13"/><span class="kw">then </span>
<font color="Maroon" title="c4">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">pp</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D11" title="SETFAM_1:def.11">SETFAM_1:def 11</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">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><span class="kw"> such that </span><br/><a NAME="E17:13"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
<font color="Maroon" title="c7">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">pp</font>
 <span class="kw">and </span><br/><a NAME="E18:13"/><span class="lab"><font color="Green" title="E18">A9</font></span>: 
<font color="Maroon" title="c7">G</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c1">T</font>
 <span class="kw">and </span><br/><a NAME="E19:13"/><span class="lab"><font color="Green" title="E19">A10</font></span>: 
<font color="Maroon" title="c7">G</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E7:13"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">G9</font> = <font color="Maroon" title="c7">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">pp</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">G9</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="c8">G9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c8">G9</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c8">G9</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )</span><br/>

<a NAME="E21:13"/>
<font color="Maroon" title="c8">G9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:13"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="xboole_1.html#T33" title="XBOOLE_1:th.33">XBOOLE_1:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:13"/>
<font color="Maroon" title="c8">G9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:13"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <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="c8">G9</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c8">G9</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )</span><br/>

<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c6">M</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:13"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="setfam_1.html#T45" title="SETFAM_1:th.45">SETFAM_1:45</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T25" title="ZFMISC_1:th.25">ZFMISC_1:25</a></span>
<br/>.= 
<font color="Maroon" title="c2">P</font>

;<br/>
<a NAME="E24:13"/><span class="kw">then </span>
<font color="Maroon" title="c2">P</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="c8">G9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#T4" title="TOPS_2:th.4">TOPS_2:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:13"/>
<font color="Maroon" title="c8">G9</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D11" title="SETFAM_1:def.11">SETFAM_1:def 11</a></span>; <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="c8">G9</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/>

<span class="kw">thus </span><a NAME="E26:13"/>
<font color="Maroon" title="c8">G9</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:13"><span class="lab"><font color="Green" title="E19">A10</font></span></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>
