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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">c</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:12"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green" title="E10">A2</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font>
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green" title="E11">A3</font></i>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</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">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">B</font></span>)</span>
 <b>and </b><br/><a NAME="E4:12"/><i><font color="Green" title="E12">A4</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font>
 ;<br/>

<a NAME="E5:12"/><i><font color="Green" title="E13">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">F</font> =  <a href="topgen_3.html#K1" title="TOPGEN_3:func.1">rng</a> <font color="Maroon" title="c2">c</font>;<br/>
<b>let </b><font color="Maroon" title="c4">T</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/>

<b>assume </b><a NAME="E6:12"/><i><font color="Green" title="E14">A6</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <span class="p1">(<span class="default"><a href="topgen_3.html#K1" title="TOPGEN_3:func.1">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> )
 ;<br/>

<a NAME="E7:12"/><i><font color="Green" title="E15">A7</font></i>: 
( <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">X</font> )
 
;<br/>
<a NAME="E8:12"/><b>then </b>
( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:12"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/>
<a NAME="E9:12"/><b>then </b>
<font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E10:12"/><b>then </b><i><font color="Green" title="E16">A8</font></i>: 
(  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_3.html#K1" title="TOPGEN_3:func.1">rng</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_3.html#K1" title="TOPGEN_3:func.1">rng</a> <font color="Maroon" title="c2">c</font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:12"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="topgen_3.html#E5:12"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="topgen_3.html#E7:12"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<div><i><font color="Green" title="E17">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:12_1"/><i><font color="Green" title="E18">A10</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_3.html#K1" title="TOPGEN_3:func.1">rng</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_3.html#K1" title="TOPGEN_3:func.1">rng</a> <font color="Maroon" title="c2">c</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c7">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:12_1"/><i><font color="Green" title="E19">A11</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:12_1"/><i><font color="Green" title="E20">A12</font></i>: 
( <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E1:12_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c8">b</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="topgen_3.html#E3:12_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="topgen_3.html#E5:12_1"><i><font color="Green" title="E20">A12</font></i></a></i>;<br/><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">B</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">B</font>
<b>by </b><i><a class="txt" href="topgen_3.html#E4:12"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="topgen_3.html#E3:12_1"><i><font color="Green" title="E19">A11</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">B</font></span>)</span>
<b>by </b><i><a class="txt" href="topgen_3.html#E4:12"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="topgen_3.html#E5:12_1"><i><font color="Green" title="E20">A12</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topgen_3.html#E3:12"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="topgen_3.html#E3:12_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="topgen_3.html#E5:12_1"><i><font color="Green" title="E20">A12</font></i></a></i>
;<br/><b>hence </b><a NAME="E8:12_1"/>
<font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_3.html#K1" title="TOPGEN_3:func.1">rng</a> <font color="Maroon" title="c2">c</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E5:12"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E18">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:12_2"/>
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font>
 ;<br/><a NAME="E2:12_2"/><b>then </b>
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/><a NAME="E3:12_2"/><b>then </b>
<font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</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">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="topgen_3.html#E3:12"><i><font color="Green" title="E11">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E4:12_2"/>
<font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T11" title="XBOOLE_1:th.11">XBOOLE_1:11</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E19">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">G</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:12_3"/><i><font color="Green" title="E20">A15</font></i>: 
<font color="Maroon" title="c5">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_3.html#K1" title="TOPGEN_3:func.1">rng</a> <font color="Maroon" title="c2">c</font>
 ;<br/><a NAME="E2:12_3"/><i><font color="Green" title="E21">A16</font></i>: 
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c5">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c5">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="topgen_3.html#E2:12"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:12_3_1"/><i><font color="Green" title="E22">A17</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c6">a</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E3:12_3_1"/><i><font color="Green" title="E23">A18</font></i>: 
 ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E1:12_3"><i><font color="Green" title="E20">A15</font></i></a>, <a class="txt" href="topgen_3.html#E1:12_3_1"><i><font color="Green" title="E22">A17</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:12_3_1"/>
<font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E12:12"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="topgen_3.html#E1:12_3_1"><i><font color="Green" title="E22">A17</font></i></a>, <a class="ref" href="mssubfam.html#T2" title="MSSUBFAM:th.2">MSSUBFAM:2</a></i>;<br/><b>hence </b><a NAME="E5:12_3_1"/>
<font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E4:12"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="topgen_3.html#E3:12_3_1"><i><font color="Green" title="E23">A18</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E4:12_3"/><b>then </b>
<font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="ref" href="mssubfam.html#T4" title="MSSUBFAM:th.4">MSSUBFAM:4</a></i>;<br/><a NAME="E5:12_3"/><b>then </b>
<font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E2:12_3"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E6:12_3"/>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_3.html#K1" title="TOPGEN_3:func.1">rng</a> <font color="Maroon" title="c2">c</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E5:12"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:12"/><i><font color="Green" title="E20">A19</font></i>: 
<font color="Maroon" title="c4">T</font> is   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>
 <b>by </b><i><a class="txt" href="topgen_3.html#E6:12"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="topgen_3.html#E10:12"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="topgen_3.html#E11:12"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="topgen_3.html#T4" target="_self" title="TOPGEN_3:th.4">Th4</a></i>;<br/>

<b>let </b><font color="Maroon" title="c5">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c7">ClA</font> =  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">A</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="topgen_3.html#E6:12"><i><font color="Green" title="E14">A6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">cB</font> = <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T</font> <b>by </b><i><a class="txt" href="topgen_3.html#E6:12"><i><font color="Green" title="E14">A6</font></i></a></i>;<br/>
<a NAME="E17:12"/>
<font color="Maroon" title="c8">cB</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_3.html#K1" title="TOPGEN_3:func.1">rng</a> <font color="Maroon" title="c2">c</font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E5:12"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E18:12"/><b>then </b>
( <font color="Maroon" title="c8">cB</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">B</font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:12"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="topgen_3.html#E6:12"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="topgen_3.html#E10:12"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="topgen_3.html#E11:12"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="topgen_3.html#E13:12"><i><font color="Green" title="E19">A14</font></i></a>, <a class="ref" href="topgen_3.html#T4" target="_self" title="TOPGEN_3:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E19:12"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T31" title="TOPS_1:th.31">TOPS_1:31</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<a NAME="E20:12"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">A</font></span>)</span>
 
;<br/>
<a NAME="E21:12"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E14:12"><i><font color="Green" title="E20">A19</font></i></a>, <a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E22:12"/><b>then </b>
<font color="Maroon" title="c7">ClA</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_3.html#K1" title="TOPGEN_3:func.1">rng</a> <font color="Maroon" title="c2">c</font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E6:12"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="topgen_3.html#E10:12"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="topgen_3.html#E11:12"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="topgen_3.html#E13:12"><i><font color="Green" title="E19">A14</font></i></a>, <a class="ref" href="topgen_3.html#T4" target="_self" title="TOPGEN_3:th.4">Th4</a></i>;<br/>
<a NAME="E23:12"/><b>then </b>
 ex <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c7">ClA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E24:12"/><b>then </b>
( <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">ClA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">ClA</font> &amp; <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">A</font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E4:12"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>
<b>hence </b><a NAME="E25:12"/>
<font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E6:12"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="topgen_3.html#E12:12"><i><font color="Green" title="E18">A13</font></i></a></i>;<br/>


</div>
